Subgroups — C₆₇×#61086 #30740337

Order 17152.

2666 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#578×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3497	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3287	C₂	1	2	yes	maximal, complemented
4	D₄×Q₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3285	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3278	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3277	C₂	1	4	yes	maximal, complemented
8	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3061	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3037	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3013	C₂	1	4	yes	maximal, complemented
12	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	4	yes	—
15	#578×C₆₇	—	2	16	no	—
16	C₂×#137×C₆₇	C₂²	1	4	yes	—
17	C₂×#137×C₆₇	C₂²	1	8	yes	—
18	C₂×#135×C₆₇	C₂²	1	4	yes	—
19	C₂×#135×C₆₇	C₂²	1	2	yes	—
20	C₂×#135×C₆₇	C₂²	1	2	yes	—
21	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
22	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#131×C₆₇	C₂²	1	2	yes	—
24	#550×C₆₇	—	2	2	no	—
25	#550×C₆₇	C₂²	1	4	yes	—
26	D₄×Q₈×C₆₇	—	2	2	no	—
27	D₄×Q₈×C₆₇	C₂²	1	4	yes	—
28	#548×C₆₇	—	2	4	no	—
29	#548×C₆₇	C₂²	1	8	yes	—
30	#537×C₆₇	—	2	4	no	—
31	#537×C₆₇	C₂²	1	4	yes	—
32	#532×C₆₇	—	2	4	no	—
33	#532×C₆₇	C₂²	1	2	yes	—
34	#532×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
36	#495×C₆₇	C₂²	1	2	yes	—
37	#495×C₆₇	C₂²	1	2	yes	—
38	#494×C₆₇	C₂²	1	1	yes	characteristic
39	#493×C₆₇	C₂²	1	1	yes	characteristic
40	#492×C₆₇	C₂²	1	2	yes	—
41	#468×C₆₇	C₂²	1	8	yes	—
42	#467×C₆₇	C₂²	1	4	yes	—
43	#466×C₆₇	C₂²	1	4	yes	—
44	#464×C₆₇	C₂²	1	8	yes	—
45	#462×C₆₇	C₂²	1	4	yes	—
46	#460×C₆₇	C₂²	1	4	yes	—
47	#452×C₆₇	C₂²	1	8	yes	—
48	#451×C₆₇	C₂²	1	8	yes	—
49	#449×C₆₇	C₂²	1	8	yes	—
50	#448×C₆₇	C₂²	1	8	yes	—
51	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
52	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
53	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
54	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
55	#436×C₆₇	C₂²	1	2	yes	—
56	#436×C₆₇	C₂²	1	2	yes	—
57	C₈×D₄×C₆₇	C₂²	1	2	yes	—
58	C₈×D₄×C₆₇	C₂²	1	2	yes	—
59	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
60	#423×C₆₇	C₂²	1	1	yes	characteristic
61	#417×C₆₇	C₂²	1	4	yes	—
62	C₂×#103×C₆₇	C₂²	1	2	yes	—
63	#408×C₆₇	C₂²	1	2	yes	—
64	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
65	2₋¹⁺⁴×C₆₇	—	2	16	no	—
66	2₋¹⁺⁴×C₆₇	—	4	8	no	—
67	C₂×#41×C₆₇	—	2	8	no	—
68	C₂×#41×C₆₇	—	2	4	no	—
69	C₂×#41×C₆₇	C₂³	1	4	yes	—
70	C₂×#41×C₆₇	C₂³	1	8	yes	—
71	C₂×#41×C₆₇	C₂³	1	2	yes	—
72	C₂²×Q₈×C₆₇	—	2	4	no	—
73	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
74	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
75	#137×C₆₇	—	2	16	no	—
76	#137×C₆₇	—	2	32	no	—
77	#135×C₆₇	—	2	16	no	—
78	#135×C₆₇	—	2	8	no	—
79	#135×C₆₇	—	4	4	no	—
80	C₂×Q₁₆×C₆₇	—	2	16	no	—
81	C₂×Q₁₆×C₆₇	D₄	1	8	yes	complemented
82	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
83	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
84	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
85	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
86	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
87	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
88	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
89	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
90	C₂×D₈×C₆₇	C₂³	1	2	yes	—
91	C₂×D₈×C₆₇	C₂³	1	2	yes	—
92	#131×C₆₇	—	2	8	no	—
93	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
94	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
95	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
96	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
97	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
98	C₄⋊Q₈×C₆₇	—	2	4	no	—
99	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
100	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
101	#127×C₆₇	—	2	4	no	—
102	#127×C₆₇	D₄	1	1	yes	characteristic
103	#127×C₆₇	C₂³	1	2	yes	—
104	#124×C₆₇	—	2	8	no	—
105	#124×C₆₇	D₄	1	2	yes	—
106	#124×C₆₇	C₂³	1	4	yes	—
107	C₂²⋊Q₈×C₆₇	—	2	4	no	—
108	C₂²⋊Q₈×C₆₇	—	2	8	no	—
109	C₂²⋊Q₈×C₆₇	—	2	8	no	—
110	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
111	#121×C₆₇	—	2	8	no	—
112	#121×C₆₇	—	2	8	no	—
113	#121×C₆₇	—	2	4	no	—
114	#121×C₆₇	—	2	2	no	—
115	#121×C₆₇	—	2	2	no	—
116	#121×C₆₇	—	2	8	no	—
117	#121×C₆₇	D₄	1	2	yes	—
118	#121×C₆₇	D₄	1	2	yes	—
119	#121×C₆₇	C₂³	1	8	yes	—
120	C₄×Q₈×C₆₇	—	2	2	no	—
121	C₄×Q₈×C₆₇	—	2	2	no	—
122	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
123	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
124	C₄×D₄×C₆₇	—	2	4	no	—
125	C₄×D₄×C₆₇	—	2	4	no	—
126	C₄×D₄×C₆₇	—	2	8	no	—
127	C₄×D₄×C₆₇	—	2	2	no	—
128	C₄×D₄×C₆₇	—	2	2	no	—
129	C₄×D₄×C₆₇	C₂³	1	2	yes	—
130	C₄×D₄×C₆₇	C₂³	1	2	yes	—
131	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
133	C₂×C₄²×C₆₇	—	2	2	no	—
134	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
135	#107×C₆₇	C₂³	1	2	yes	—
136	#107×C₆₇	C₂³	1	2	yes	—
137	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
138	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
139	#103×C₆₇	C₂³	1	4	yes	—
140	#103×C₆₇	C₂³	1	8	yes	—
141	#103×C₆₇	C₂³	1	4	yes	—
142	#102×C₆₇	C₂³	1	8	yes	—
143	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
144	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
145	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
146	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
147	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
148	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
149	#41×C₆₇	—	2	4	no	—
150	#41×C₆₇	—	2	4	no	—
151	#41×C₆₇	—	2	16	no	—
152	#41×C₆₇	—	2	8	no	—
153	#41×C₆₇	—	2	8	no	—
154	#41×C₆₇	—	4	8	no	—
155	#41×C₆₇	—	4	32	no	—
156	#41×C₆₇	—	4	16	no	—
157	C₂×Q₈×C₆₇	—	2	2	no	—
158	C₂×Q₈×C₆₇	—	2	8	no	—
159	C₂×Q₈×C₆₇	—	2	4	no	—
160	C₂×Q₈×C₆₇	—	2	16	no	—
161	C₂×Q₈×C₆₇	—	4	8	no	—
162	C₂×Q₈×C₆₇	—	4	16	no	—
163	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
164	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
165	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
166	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
167	C₂×D₄×C₆₇	—	2	4	no	—
168	C₂×D₄×C₆₇	—	2	8	no	—
169	C₂×D₄×C₆₇	—	2	8	no	—
170	C₂×D₄×C₆₇	—	2	2	no	—
171	C₂×D₄×C₆₇	—	2	2	no	—
172	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
173	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
174	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
175	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
176	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₆₇	—	2	2	no	—
178	C₂²×C₄×C₆₇	—	2	8	no	—
179	C₂²×C₄×C₆₇	—	2	2	no	—
180	C₂²×C₄×C₆₇	—	2	8	no	—
181	C₂²×C₄×C₆₇	—	2	4	no	—
182	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
183	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
185	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
186	Q₁₆×C₆₇	—	2	16	no	—
187	Q₁₆×C₆₇	—	2	4	no	—
188	Q₁₆×C₆₇	—	4	2	no	—
189	Q₁₆×C₆₇	C₂×D₄	1	8	yes	—
190	Q₁₆×C₆₇	C₂×D₄	1	8	yes	—
191	SD₁₆×C₆₇	—	2	16	no	—
192	SD₁₆×C₆₇	—	2	8	no	—
193	SD₁₆×C₆₇	—	4	4	no	—
194	D₈×C₆₇	—	2	4	no	—
195	D₈×C₆₇	—	4	2	no	—
196	M₁₆×C₆₇	—	2	4	no	—
197	M₁₆×C₆₇	—	2	8	no	—
198	C₂×C₈×C₆₇	—	2	2	no	—
199	C₂×C₈×C₆₇	—	2	8	no	—
200	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
201	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
202	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
203	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
204	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
205	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
206	C₄⋊C₄×C₆₇	—	2	4	no	—
207	C₄⋊C₄×C₆₇	—	2	4	no	—
208	C₄⋊C₄×C₆₇	—	2	2	no	—
209	C₄⋊C₄×C₆₇	—	2	2	no	—
210	C₄⋊C₄×C₆₇	—	2	2	no	—
211	C₄⋊C₄×C₆₇	—	2	8	no	—
212	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
213	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
214	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
215	C₂²⋊C₄×C₆₇	—	2	8	no	—
216	C₂²⋊C₄×C₆₇	—	2	4	no	—
217	C₂²⋊C₄×C₆₇	—	2	4	no	—
218	C₂²⋊C₄×C₆₇	—	2	8	no	—
219	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
220	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
221	C₄²×C₆₇	—	2	4	no	—
222	C₄²×C₆₇	—	2	4	no	—
223	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
224	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
225	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
226	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
227	C₂³×C₆₇	—	2	4	no	—
228	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
229	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
230	Q₈×C₆₇	—	2	8	no	—
231	Q₈×C₆₇	—	2	4	no	—
232	Q₈×C₆₇	—	2	4	no	—
233	Q₈×C₆₇	—	2	4	no	—
234	Q₈×C₆₇	—	2	4	no	—
235	Q₈×C₆₇	—	4	16	no	—
236	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
237	D₄×C₆₇	—	2	8	no	—
238	D₄×C₆₇	—	2	4	no	—
239	D₄×C₆₇	—	2	4	no	—
240	D₄×C₆₇	—	2	4	no	—
241	D₄×C₆₇	—	2	4	no	—
242	D₄×C₆₇	—	4	2	no	—
243	D₄×C₆₇	—	4	16	no	—
244	D₄×C₆₇	—	4	8	no	—
245	D₄×C₆₇	—	4	4	no	—
246	D₄×C₆₇	—	8	2	no	—
247	D₄×C₆₇	—	8	8	no	—
248	C₂×C₄×C₆₇	—	2	8	no	—
249	C₂×C₄×C₆₇	—	2	4	no	—
250	C₂×C₄×C₆₇	—	2	2	no	—
251	C₂×C₄×C₆₇	—	2	4	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	2	4	no	—
255	C₂×C₄×C₆₇	—	4	1	no	—
256	C₂×C₄×C₆₇	—	4	16	no	—
257	C₂×C₄×C₆₇	—	4	16	no	—
258	C₂×C₄×C₆₇	—	4	8	no	—
259	C₂×C₄×C₆₇	—	8	2	no	—
260	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
264	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
266	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
268	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
269	C₈×C₆₇	—	2	4	no	—
270	C₈×C₆₇	—	2	1	no	—
271	C₈×C₆₇	—	2	1	no	—
272	C₈×C₆₇	C₂²×D₄	1	2	yes	—
273	C₈×C₆₇	C₂²×D₄	1	2	yes	—
274	C₂²×C₆₇	—	2	1	no	—
275	C₂²×C₆₇	—	2	1	no	—
276	C₂²×C₆₇	—	2	4	no	—
277	C₂²×C₆₇	—	2	4	no	—
278	C₂²×C₆₇	—	2	4	no	—
279	C₂²×C₆₇	—	4	4	no	—
280	C₂²×C₆₇	—	4	4	no	—
281	C₂²×C₆₇	—	4	4	no	—
282	C₂²×C₆₇	D₄²	1	2	yes	—
283	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
284	C₄×C₆₇	—	2	1	no	—
285	C₄×C₆₇	—	2	1	no	—
286	C₄×C₆₇	—	2	8	no	—
287	C₄×C₆₇	—	2	4	no	—
288	C₄×C₆₇	—	2	1	no	—
289	C₄×C₆₇	—	4	4	no	—
290	C₄×C₆₇	—	4	4	no	—
291	C₄×C₆₇	D₄²	1	2	yes	—
292	C₄×C₆₇	#578	1	2	yes	complemented
293	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
294	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
295	#61086	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
296	C₂×C₆₇	—	2	2	no	—
297	C₂×C₆₇	—	4	4	no	—
298	C₂×C₆₇	—	8	4	no	—
299	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
300	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
301	C₂×C₆₇	#3584	1	1	yes	characteristic
302	C₂×#578	C₂×C₆₇	1	2	yes	complemented
303	#3497	C₂×C₆₇	1	2	yes	complemented
304	#3287	C₂×C₆₇	1	2	yes	complemented
305	D₄×Q₁₆	C₂×C₆₇	1	2	yes	complemented
306	#3285	C₂×C₆₇	1	4	yes	complemented
307	#3278	C₂×C₆₇	1	4	yes	complemented
308	#3277	C₂×C₆₇	1	4	yes	complemented
309	#3147	C₂×C₆₇	1	1	yes	characteristic, complemented
310	#3061	C₂×C₆₇	1	2	yes	complemented
311	#3037	C₂×C₆₇	1	2	yes	complemented
312	#3013	C₂×C₆₇	1	4	yes	complemented
313	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
314	#2931	C₂×C₆₇	1	1	yes	characteristic, complemented
315	C₆₇	#61086	1	1	yes	characteristic, direct product factor, Sylow
316	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	4	yes	—
317	#578	—	2	16	no	—
318	C₂×#137	C₂²×C₆₇	1	4	yes	—
319	C₂×#137	C₂²×C₆₇	1	8	yes	—
320	C₂×#135	C₂²×C₆₇	1	2	yes	—
321	C₂×#135	C₂²×C₆₇	1	4	yes	—
322	C₂×#135	C₂²×C₆₇	1	2	yes	—
323	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	—
324	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
325	C₂×#131	C₂²×C₆₇	1	2	yes	—
326	#550	—	2	2	no	—
327	#550	C₂²×C₆₇	1	4	yes	—
328	D₄×Q₈	—	2	2	no	—
329	D₄×Q₈	C₂²×C₆₇	1	4	yes	—
330	#548	—	2	4	no	—
331	#548	C₂²×C₆₇	1	8	yes	—
332	#537	—	2	4	no	—
333	#537	C₂²×C₆₇	1	4	yes	—
334	#532	—	2	4	no	—
335	#532	C₂²×C₆₇	1	2	yes	—
336	#532	C₂²×C₆₇	1	1	yes	characteristic
337	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
338	#495	C₂²×C₆₇	1	2	yes	—
339	#495	C₂²×C₆₇	1	2	yes	—
340	#494	C₂²×C₆₇	1	1	yes	characteristic
341	#493	C₂²×C₆₇	1	1	yes	characteristic
342	#492	C₂²×C₆₇	1	2	yes	—
343	#468	C₂²×C₆₇	1	8	yes	—
344	#467	C₂²×C₆₇	1	4	yes	—
345	#466	C₂²×C₆₇	1	4	yes	—
346	#464	C₂²×C₆₇	1	8	yes	—
347	#462	C₂²×C₆₇	1	4	yes	—
348	#460	C₂²×C₆₇	1	4	yes	—
349	#452	C₂²×C₆₇	1	8	yes	—
350	#451	C₂²×C₆₇	1	8	yes	—
351	#449	C₂²×C₆₇	1	8	yes	—
352	#448	C₂²×C₆₇	1	8	yes	—
353	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
354	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
355	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
356	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
357	#436	C₂²×C₆₇	1	2	yes	—
358	#436	C₂²×C₆₇	1	2	yes	—
359	C₈×D₄	C₂²×C₆₇	1	2	yes	—
360	C₈×D₄	C₂²×C₆₇	1	2	yes	—
361	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
362	#423	C₂²×C₆₇	1	1	yes	characteristic
363	#417	C₂²×C₆₇	1	4	yes	—
364	C₂×#103	C₂²×C₆₇	1	2	yes	—
365	#408	C₂²×C₆₇	1	2	yes	—
366	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
367	2₋¹⁺⁴	—	2	16	no	—
368	2₋¹⁺⁴	—	4	8	no	—
369	C₂×#41	—	2	8	no	—
370	C₂×#41	—	2	4	no	—
371	C₂×#41	C₂³×C₆₇	1	4	yes	—
372	C₂×#41	C₂³×C₆₇	1	8	yes	—
373	C₂×#41	C₂³×C₆₇	1	2	yes	—
374	C₂²×Q₈	—	2	4	no	—
375	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
376	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
377	#137	—	2	16	no	—
378	#137	—	2	32	no	—
379	#135	—	2	16	no	—
380	#135	—	2	8	no	—
381	#135	—	4	4	no	—
382	C₂×Q₁₆	—	2	16	no	—
383	C₂×Q₁₆	D₄×C₆₇	1	8	yes	complemented
384	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
385	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
386	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
387	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
388	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
389	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
390	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
391	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
392	C₂×D₈	C₂³×C₆₇	1	2	yes	—
393	C₂×D₈	C₂³×C₆₇	1	2	yes	—
394	#131	—	2	8	no	—
395	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
396	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
397	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
398	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
399	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
400	C₄⋊Q₈	—	2	4	no	—
401	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
402	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
403	#127	—	2	4	no	—
404	#127	D₄×C₆₇	1	1	yes	characteristic
405	#127	C₂³×C₆₇	1	2	yes	—
406	#124	—	2	8	no	—
407	#124	D₄×C₆₇	1	2	yes	—
408	#124	C₂³×C₆₇	1	4	yes	—
409	C₂²⋊Q₈	—	2	4	no	—
410	C₂²⋊Q₈	—	2	8	no	—
411	C₂²⋊Q₈	—	2	8	no	—
412	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
413	#121	—	2	8	no	—
414	#121	—	2	8	no	—
415	#121	—	2	4	no	—
416	#121	—	2	2	no	—
417	#121	—	2	2	no	—
418	#121	—	2	8	no	—
419	#121	D₄×C₆₇	1	2	yes	—
420	#121	D₄×C₆₇	1	2	yes	—
421	#121	C₂³×C₆₇	1	8	yes	—
422	C₄×Q₈	—	2	2	no	—
423	C₄×Q₈	—	2	2	no	—
424	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
425	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
426	C₄×D₄	—	2	4	no	—
427	C₄×D₄	—	2	4	no	—
428	C₄×D₄	—	2	8	no	—
429	C₄×D₄	—	2	2	no	—
430	C₄×D₄	—	2	2	no	—
431	C₄×D₄	C₂³×C₆₇	1	2	yes	—
432	C₄×D₄	C₂³×C₆₇	1	2	yes	—
433	C₂×C₄⋊C₄	—	2	2	no	—
434	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
435	C₂×C₄²	—	2	2	no	—
436	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
437	#107	C₂³×C₆₇	1	2	yes	—
438	#107	C₂³×C₆₇	1	2	yes	—
439	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
440	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
441	#103	C₂³×C₆₇	1	4	yes	—
442	#103	C₂³×C₆₇	1	4	yes	—
443	#103	C₂³×C₆₇	1	8	yes	—
444	#102	C₂³×C₆₇	1	8	yes	—
445	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
446	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
447	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
448	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
449	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
450	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
451	#41	—	2	4	no	—
452	#41	—	2	4	no	—
453	#41	—	2	16	no	—
454	#41	—	2	8	no	—
455	#41	—	2	8	no	—
456	#41	—	4	8	no	—
457	#41	—	4	32	no	—
458	#41	—	4	16	no	—
459	C₂×Q₈	—	2	2	no	—
460	C₂×Q₈	—	2	8	no	—
461	C₂×Q₈	—	2	4	no	—
462	C₂×Q₈	—	2	16	no	—
463	C₂×Q₈	—	4	8	no	—
464	C₂×Q₈	—	4	16	no	—
465	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
466	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
467	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
468	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	2	8	no	—
471	C₂×D₄	—	2	8	no	—
472	C₂×D₄	—	2	2	no	—
473	C₂×D₄	—	2	2	no	—
474	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
475	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
476	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
477	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
478	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	8	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	8	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
485	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
486	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
487	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
488	Q₁₆	—	2	16	no	—
489	Q₁₆	—	2	4	no	—
490	Q₁₆	—	4	2	no	—
491	Q₁₆	C₂×D₄×C₆₇	1	8	yes	—
492	Q₁₆	C₂×D₄×C₆₇	1	8	yes	—
493	SD₁₆	—	2	16	no	—
494	SD₁₆	—	2	8	no	—
495	SD₁₆	—	4	4	no	—
496	D₈	—	2	4	no	—
497	D₈	—	4	2	no	—
498	M₁₆	—	2	4	no	—
499	M₁₆	—	2	8	no	—
500	C₂×C₈	—	2	2	no	—
501	C₂×C₈	—	2	8	no	—
502	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
503	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
504	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
505	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
506	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
507	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	—	2	4	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	8	no	—
514	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
515	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
516	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
517	C₂²⋊C₄	—	2	8	no	—
518	C₂²⋊C₄	—	2	4	no	—
519	C₂²⋊C₄	—	2	4	no	—
520	C₂²⋊C₄	—	2	8	no	—
521	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
522	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
523	C₄²	—	2	4	no	—
524	C₄²	—	2	4	no	—
525	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
526	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
527	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
528	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
529	C₂³	—	2	4	no	—
530	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
531	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
532	Q₈	—	2	8	no	—
533	Q₈	—	2	4	no	—
534	Q₈	—	2	4	no	—
535	Q₈	—	2	4	no	—
536	Q₈	—	2	4	no	—
537	Q₈	—	4	16	no	—
538	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
539	D₄	—	2	8	no	—
540	D₄	—	2	4	no	—
541	D₄	—	2	4	no	—
542	D₄	—	2	4	no	—
543	D₄	—	2	4	no	—
544	D₄	—	4	2	no	—
545	D₄	—	4	16	no	—
546	D₄	—	4	8	no	—
547	D₄	—	4	4	no	—
548	D₄	—	8	2	no	—
549	D₄	—	8	8	no	—
550	C₂×C₄	—	2	8	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	16	no	—
559	C₂×C₄	—	4	16	no	—
560	C₂×C₄	—	4	8	no	—
561	C₂×C₄	—	8	2	no	—
562	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
563	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
564	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
565	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
566	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
567	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
568	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
569	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
570	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
571	C₈	—	2	4	no	—
572	C₈	—	2	1	no	—
573	C₈	—	2	1	no	—
574	C₈	C₂²×D₄×C₆₇	1	2	yes	—
575	C₈	C₂²×D₄×C₆₇	1	2	yes	—
576	C₂²	—	2	1	no	—
577	C₂²	—	2	1	no	—
578	C₂²	—	2	4	no	—
579	C₂²	—	2	4	no	—
580	C₂²	—	2	4	no	—
581	C₂²	—	4	4	no	—
582	C₂²	—	4	4	no	—
583	C₂²	—	4	4	no	—
584	C₂²	D₄²×C₆₇	1	2	yes	—
585	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
586	C₄	—	2	1	no	—
587	C₄	—	2	1	no	—
588	C₄	—	2	1	no	—
589	C₄	—	2	8	no	—
590	C₄	—	2	4	no	—
591	C₄	—	4	4	no	—
592	C₄	—	4	4	no	—
593	C₄	D₄²×C₆₇	1	2	yes	—
594	C₄	#578×C₆₇	1	2	yes	complemented
595	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
596	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
597	C₂	—	2	2	no	—
598	C₂	—	4	4	no	—
599	C₂	—	8	4	no	—
600	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
601	C₂	C₂×#578×C₆₇	1	1	yes	characteristic
602	C₂	C₆₇×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #532×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#578×C₆₇	C₂	semidirect
2. C₆₇×#3497	C₂	semidirect
3. C₆₇×#3287	C₂	semidirect
4. D₄×Q₁₆×C₆₇	C₂	semidirect
5. C₆₇×#3285	C₂	semidirect
6. C₆₇×#3278	C₂	semidirect
7. C₆₇×#3277	C₂	semidirect
8. C₆₇×#3147	C₂	semidirect
9. C₆₇×#3061	C₂	semidirect
10. C₆₇×#3037	C₂	semidirect
11. C₆₇×#3013	C₂	semidirect
12. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
13. C₆₇×#2931	C₂	semidirect
81. C₂×Q₁₆×C₆₇	D₄	semidirect
292. C₄×C₆₇	#578	semidirect
295. #61086	C₆₇	direct (N × H)
302. C₂×#578	C₂×C₆₇	semidirect
303. #3497	C₂×C₆₇	semidirect
304. #3287	C₂×C₆₇	semidirect
305. D₄×Q₁₆	C₂×C₆₇	semidirect
306. #3285	C₂×C₆₇	semidirect
307. #3278	C₂×C₆₇	semidirect
308. #3277	C₂×C₆₇	semidirect
309. #3147	C₂×C₆₇	semidirect
310. #3061	C₂×C₆₇	semidirect
311. #3037	C₂×C₆₇	semidirect
312. #3013	C₂×C₆₇	semidirect
313. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
314. #2931	C₂×C₆₇	semidirect
315. C₆₇	#61086	direct (N × H)
383. C₂×Q₁₆	D₄×C₆₇	semidirect
594. C₄	#578×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#578×C₆₇	2
C₆₇×#3497	2
C₆₇×#3287	2
D₄×Q₁₆×C₆₇	2
C₆₇×#3285	2
C₆₇×#3278	2
C₆₇×#3277	2
C₆₇×#3147	2
C₆₇×#3061	2
C₆₇×#3037	2
C₆₇×#3013	2
C₂×C₄×Q₁₆×C₆₇	2
C₆₇×#2931	2
#61086	67

Subgroups — C67×#61086 #30740337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61086 #30740337