Subgroups — C₆₇×#61081 #30740332

Order 17152.

2658 conjugacy classes of subgroups.

The subgroup lattice has 522 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×#3271	C₂	1	8	yes	maximal, complemented
4	C₆₇×#3270	C₂	1	8	yes	maximal, complemented
5	C₆₇×#3150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#3060	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3037	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3013	C₂	1	4	yes	maximal, complemented
9	C₂×#441×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2932	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	4	yes	—
12	#578×C₆₇	—	2	16	no	—
13	C₂×#137×C₆₇	C₂²	1	4	yes	—
14	C₂×#137×C₆₇	C₂²	1	8	yes	—
15	C₂×#135×C₆₇	C₂²	1	4	yes	—
16	C₂×#135×C₆₇	C₂²	1	2	yes	—
17	C₂×#135×C₆₇	C₂²	1	2	yes	—
18	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
19	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#131×C₆₇	C₂²	1	2	yes	—
21	#550×C₆₇	—	2	2	no	—
22	#550×C₆₇	C₂²	1	4	yes	—
23	D₄×Q₈×C₆₇	—	2	2	no	—
24	D₄×Q₈×C₆₇	C₂²	1	4	yes	—
25	#548×C₆₇	—	2	4	no	—
26	#548×C₆₇	C₂²	1	8	yes	—
27	#537×C₆₇	—	2	4	no	—
28	#537×C₆₇	C₂²	1	4	yes	—
29	#532×C₆₇	—	2	4	no	—
30	#532×C₆₇	C₂²	1	2	yes	—
31	#532×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
33	#497×C₆₇	C₂²	1	4	yes	—
34	#496×C₆₇	C₂²	1	4	yes	—
35	#470×C₆₇	C₂²	1	4	yes	—
36	#469×C₆₇	C₂²	1	4	yes	—
37	#465×C₆₇	C₂²	1	8	yes	—
38	#464×C₆₇	C₂²	1	8	yes	—
39	#462×C₆₇	C₂²	1	4	yes	—
40	#460×C₆₇	C₂²	1	4	yes	—
41	#452×C₆₇	C₂²	1	8	yes	—
42	#451×C₆₇	C₂²	1	8	yes	—
43	#449×C₆₇	C₂²	1	8	yes	—
44	#448×C₆₇	C₂²	1	8	yes	—
45	#441×C₆₇	C₂²	1	8	yes	—
46	#435×C₆₇	C₂²	1	8	yes	—
47	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
48	#423×C₆₇	C₂²	1	1	yes	characteristic
49	#417×C₆₇	C₂²	1	4	yes	—
50	C₂×#103×C₆₇	C₂²	1	2	yes	—
51	#408×C₆₇	C₂²	1	2	yes	—
52	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
53	2₋¹⁺⁴×C₆₇	—	2	16	no	—
54	2₋¹⁺⁴×C₆₇	—	4	8	no	—
55	C₂×#41×C₆₇	—	2	8	no	—
56	C₂×#41×C₆₇	—	2	4	no	—
57	C₂×#41×C₆₇	C₂³	1	4	yes	—
58	C₂×#41×C₆₇	C₂³	1	8	yes	—
59	C₂×#41×C₆₇	C₂³	1	2	yes	—
60	C₂²×Q₈×C₆₇	—	2	4	no	—
61	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
62	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
63	#137×C₆₇	—	2	16	no	—
64	#137×C₆₇	—	2	32	no	—
65	#135×C₆₇	—	2	16	no	—
66	#135×C₆₇	—	2	8	no	—
67	#135×C₆₇	—	4	4	no	—
68	C₂×Q₁₆×C₆₇	—	2	16	no	—
69	C₂×Q₁₆×C₆₇	D₄	1	8	yes	complemented
70	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
71	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
72	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
73	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
74	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
75	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
76	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
77	C₂×D₈×C₆₇	C₂³	1	2	yes	—
78	C₂×D₈×C₆₇	C₂³	1	2	yes	—
79	#131×C₆₇	—	2	8	no	—
80	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
81	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
82	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
83	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₄⋊Q₈×C₆₇	—	2	4	no	—
86	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
87	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
88	#127×C₆₇	—	2	4	no	—
89	#127×C₆₇	D₄	1	1	yes	characteristic
90	#127×C₆₇	C₂³	1	2	yes	—
91	#124×C₆₇	—	2	8	no	—
92	#124×C₆₇	D₄	1	2	yes	—
93	#124×C₆₇	C₂³	1	4	yes	—
94	C₂²⋊Q₈×C₆₇	—	2	4	no	—
95	C₂²⋊Q₈×C₆₇	—	2	8	no	—
96	C₂²⋊Q₈×C₆₇	—	2	8	no	—
97	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
98	#121×C₆₇	—	2	8	no	—
99	#121×C₆₇	—	2	8	no	—
100	#121×C₆₇	—	2	4	no	—
101	#121×C₆₇	—	2	2	no	—
102	#121×C₆₇	—	2	2	no	—
103	#121×C₆₇	—	2	8	no	—
104	#121×C₆₇	D₄	1	4	yes	—
105	#121×C₆₇	C₂³	1	8	yes	—
106	C₄×Q₈×C₆₇	—	2	2	no	—
107	C₄×Q₈×C₆₇	—	2	2	no	—
108	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
109	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
110	C₄×D₄×C₆₇	—	2	4	no	—
111	C₄×D₄×C₆₇	—	2	4	no	—
112	C₄×D₄×C₆₇	—	2	8	no	—
113	C₄×D₄×C₆₇	—	2	2	no	—
114	C₄×D₄×C₆₇	—	2	2	no	—
115	C₄×D₄×C₆₇	C₂³	1	4	yes	—
116	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
118	C₂×C₄²×C₆₇	—	2	2	no	—
119	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
120	#106×C₆₇	C₂³	1	4	yes	—
121	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
122	#103×C₆₇	C₂³	1	8	yes	—
123	#103×C₆₇	C₂³	1	8	yes	—
124	#102×C₆₇	C₂³	1	8	yes	—
125	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
126	#97×C₆₇	C₂³	1	4	yes	—
127	#41×C₆₇	—	2	4	no	—
128	#41×C₆₇	—	2	4	no	—
129	#41×C₆₇	—	2	16	no	—
130	#41×C₆₇	—	2	8	no	—
131	#41×C₆₇	—	2	8	no	—
132	#41×C₆₇	—	4	8	no	—
133	#41×C₆₇	—	4	32	no	—
134	#41×C₆₇	—	4	16	no	—
135	C₂×Q₈×C₆₇	—	2	2	no	—
136	C₂×Q₈×C₆₇	—	2	8	no	—
137	C₂×Q₈×C₆₇	—	2	4	no	—
138	C₂×Q₈×C₆₇	—	2	16	no	—
139	C₂×Q₈×C₆₇	—	4	8	no	—
140	C₂×Q₈×C₆₇	—	4	16	no	—
141	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
142	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
143	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
144	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
145	C₂×D₄×C₆₇	—	2	4	no	—
146	C₂×D₄×C₆₇	—	2	8	no	—
147	C₂×D₄×C₆₇	—	2	8	no	—
148	C₂×D₄×C₆₇	—	2	2	no	—
149	C₂×D₄×C₆₇	—	2	2	no	—
150	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
151	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
152	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
153	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₆₇	—	2	2	no	—
155	C₂²×C₄×C₆₇	—	2	8	no	—
156	C₂²×C₄×C₆₇	—	2	2	no	—
157	C₂²×C₄×C₆₇	—	2	8	no	—
158	C₂²×C₄×C₆₇	—	2	4	no	—
159	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
162	Q₁₆×C₆₇	—	2	16	no	—
163	Q₁₆×C₆₇	—	2	4	no	—
164	Q₁₆×C₆₇	—	4	2	no	—
165	Q₁₆×C₆₇	C₂×D₄	1	16	yes	—
166	SD₁₆×C₆₇	—	2	16	no	—
167	SD₁₆×C₆₇	—	2	8	no	—
168	SD₁₆×C₆₇	—	4	4	no	—
169	D₈×C₆₇	—	2	4	no	—
170	D₈×C₆₇	—	4	2	no	—
171	M₁₆×C₆₇	—	2	4	no	—
172	M₁₆×C₆₇	—	2	8	no	—
173	C₂×C₈×C₆₇	—	2	2	no	—
174	C₂×C₈×C₆₇	—	2	8	no	—
175	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
176	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
177	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
178	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
179	C₄⋊C₄×C₆₇	—	2	4	no	—
180	C₄⋊C₄×C₆₇	—	2	4	no	—
181	C₄⋊C₄×C₆₇	—	2	2	no	—
182	C₄⋊C₄×C₆₇	—	2	2	no	—
183	C₄⋊C₄×C₆₇	—	2	2	no	—
184	C₄⋊C₄×C₆₇	—	2	8	no	—
185	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
186	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
187	C₂²⋊C₄×C₆₇	—	2	8	no	—
188	C₂²⋊C₄×C₆₇	—	2	4	no	—
189	C₂²⋊C₄×C₆₇	—	2	4	no	—
190	C₂²⋊C₄×C₆₇	—	2	8	no	—
191	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
192	C₄²×C₆₇	—	2	4	no	—
193	C₄²×C₆₇	—	2	4	no	—
194	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
196	C₄²×C₆₇	C₂⁴	1	2	yes	—
197	C₂³×C₆₇	—	2	4	no	—
198	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
199	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
200	Q₈×C₆₇	—	2	8	no	—
201	Q₈×C₆₇	—	2	4	no	—
202	Q₈×C₆₇	—	2	4	no	—
203	Q₈×C₆₇	—	2	4	no	—
204	Q₈×C₆₇	—	2	4	no	—
205	Q₈×C₆₇	—	4	16	no	—
206	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
207	D₄×C₆₇	—	2	8	no	—
208	D₄×C₆₇	—	2	4	no	—
209	D₄×C₆₇	—	2	4	no	—
210	D₄×C₆₇	—	2	4	no	—
211	D₄×C₆₇	—	4	4	no	—
212	D₄×C₆₇	—	4	16	no	—
213	D₄×C₆₇	—	4	4	no	—
214	D₄×C₆₇	—	4	8	no	—
215	D₄×C₆₇	—	8	2	no	—
216	D₄×C₆₇	—	8	8	no	—
217	C₂×C₄×C₆₇	—	2	8	no	—
218	C₂×C₄×C₆₇	—	2	4	no	—
219	C₂×C₄×C₆₇	—	2	4	no	—
220	C₂×C₄×C₆₇	—	2	4	no	—
221	C₂×C₄×C₆₇	—	2	4	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	4	2	no	—
224	C₂×C₄×C₆₇	—	4	16	no	—
225	C₂×C₄×C₆₇	—	4	16	no	—
226	C₂×C₄×C₆₇	—	4	8	no	—
227	C₂×C₄×C₆₇	—	8	2	no	—
228	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
229	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
230	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
231	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
233	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
234	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
235	C₈×C₆₇	—	2	4	no	—
236	C₈×C₆₇	—	2	2	no	—
237	C₈×C₆₇	C₂²×D₄	1	4	yes	—
238	C₂²×C₆₇	—	2	2	no	—
239	C₂²×C₆₇	—	2	8	no	—
240	C₂²×C₆₇	—	2	4	no	—
241	C₂²×C₆₇	—	4	4	no	—
242	C₂²×C₆₇	—	4	4	no	—
243	C₂²×C₆₇	—	4	4	no	—
244	C₂²×C₆₇	D₄²	1	2	yes	—
245	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
246	C₄×C₆₇	—	2	2	no	—
247	C₄×C₆₇	—	2	2	no	—
248	C₄×C₆₇	—	2	8	no	—
249	C₄×C₆₇	—	2	4	no	—
250	C₄×C₆₇	—	4	4	no	—
251	C₄×C₆₇	—	4	4	no	—
252	C₄×C₆₇	D₄²	1	2	yes	—
253	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
254	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
255	#61081	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
256	C₂×C₆₇	—	2	2	no	—
257	C₂×C₆₇	—	4	4	no	—
258	C₂×C₆₇	—	8	4	no	—
259	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
260	C₂×C₆₇	#3586	1	2	yes	—
261	C₂×#578	C₂×C₆₇	1	2	yes	complemented
262	#3497	C₂×C₆₇	1	2	yes	complemented
263	#3271	C₂×C₆₇	1	8	yes	complemented
264	#3270	C₂×C₆₇	1	8	yes	complemented
265	#3150	C₂×C₆₇	1	1	yes	characteristic, complemented
266	#3060	C₂×C₆₇	1	2	yes	complemented
267	#3037	C₂×C₆₇	1	2	yes	complemented
268	#3013	C₂×C₆₇	1	4	yes	complemented
269	C₂×#441	C₂×C₆₇	1	1	yes	characteristic, complemented
270	#2932	C₂×C₆₇	1	1	yes	characteristic, complemented
271	C₆₇	#61081	1	1	yes	characteristic, direct product factor, Sylow
272	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	4	yes	—
273	#578	—	2	16	no	—
274	C₂×#137	C₂²×C₆₇	1	4	yes	—
275	C₂×#137	C₂²×C₆₇	1	8	yes	—
276	C₂×#135	C₂²×C₆₇	1	2	yes	—
277	C₂×#135	C₂²×C₆₇	1	4	yes	—
278	C₂×#135	C₂²×C₆₇	1	2	yes	—
279	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	—
280	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
281	C₂×#131	C₂²×C₆₇	1	2	yes	—
282	#550	—	2	2	no	—
283	#550	C₂²×C₆₇	1	4	yes	—
284	D₄×Q₈	—	2	2	no	—
285	D₄×Q₈	C₂²×C₆₇	1	4	yes	—
286	#548	—	2	4	no	—
287	#548	C₂²×C₆₇	1	8	yes	—
288	#537	—	2	4	no	—
289	#537	C₂²×C₆₇	1	4	yes	—
290	#532	—	2	4	no	—
291	#532	C₂²×C₆₇	1	2	yes	—
292	#532	C₂²×C₆₇	1	1	yes	characteristic
293	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
294	#497	C₂²×C₆₇	1	4	yes	—
295	#496	C₂²×C₆₇	1	4	yes	—
296	#470	C₂²×C₆₇	1	4	yes	—
297	#469	C₂²×C₆₇	1	4	yes	—
298	#465	C₂²×C₆₇	1	8	yes	—
299	#464	C₂²×C₆₇	1	8	yes	—
300	#462	C₂²×C₆₇	1	4	yes	—
301	#460	C₂²×C₆₇	1	4	yes	—
302	#452	C₂²×C₆₇	1	8	yes	—
303	#451	C₂²×C₆₇	1	8	yes	—
304	#449	C₂²×C₆₇	1	8	yes	—
305	#448	C₂²×C₆₇	1	8	yes	—
306	#441	C₂²×C₆₇	1	8	yes	—
307	#435	C₂²×C₆₇	1	8	yes	—
308	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
309	#423	C₂²×C₆₇	1	1	yes	characteristic
310	#417	C₂²×C₆₇	1	4	yes	—
311	C₂×#103	C₂²×C₆₇	1	2	yes	—
312	#408	C₂²×C₆₇	1	2	yes	—
313	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
314	2₋¹⁺⁴	—	2	16	no	—
315	2₋¹⁺⁴	—	4	8	no	—
316	C₂×#41	—	2	8	no	—
317	C₂×#41	—	2	4	no	—
318	C₂×#41	C₂³×C₆₇	1	4	yes	—
319	C₂×#41	C₂³×C₆₇	1	8	yes	—
320	C₂×#41	C₂³×C₆₇	1	2	yes	—
321	C₂²×Q₈	—	2	4	no	—
322	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
323	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
324	#137	—	2	16	no	—
325	#137	—	2	32	no	—
326	#135	—	2	16	no	—
327	#135	—	2	8	no	—
328	#135	—	4	4	no	—
329	C₂×Q₁₆	—	2	16	no	—
330	C₂×Q₁₆	D₄×C₆₇	1	8	yes	complemented
331	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
332	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
333	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
334	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
335	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
336	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
337	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
338	C₂×D₈	C₂³×C₆₇	1	2	yes	—
339	C₂×D₈	C₂³×C₆₇	1	2	yes	—
340	#131	—	2	8	no	—
341	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
342	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
343	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
344	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
346	C₄⋊Q₈	—	2	4	no	—
347	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
348	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
349	#127	—	2	4	no	—
350	#127	D₄×C₆₇	1	1	yes	characteristic
351	#127	C₂³×C₆₇	1	2	yes	—
352	#124	—	2	8	no	—
353	#124	D₄×C₆₇	1	2	yes	—
354	#124	C₂³×C₆₇	1	4	yes	—
355	C₂²⋊Q₈	—	2	4	no	—
356	C₂²⋊Q₈	—	2	8	no	—
357	C₂²⋊Q₈	—	2	8	no	—
358	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
359	#121	—	2	8	no	—
360	#121	—	2	8	no	—
361	#121	—	2	4	no	—
362	#121	—	2	2	no	—
363	#121	—	2	2	no	—
364	#121	—	2	8	no	—
365	#121	D₄×C₆₇	1	4	yes	—
366	#121	C₂³×C₆₇	1	8	yes	—
367	C₄×Q₈	—	2	2	no	—
368	C₄×Q₈	—	2	2	no	—
369	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
370	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
371	C₄×D₄	—	2	4	no	—
372	C₄×D₄	—	2	4	no	—
373	C₄×D₄	—	2	8	no	—
374	C₄×D₄	—	2	2	no	—
375	C₄×D₄	—	2	2	no	—
376	C₄×D₄	C₂³×C₆₇	1	4	yes	—
377	C₂×C₄⋊C₄	—	2	2	no	—
378	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
379	C₂×C₄²	—	2	2	no	—
380	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
381	#106	C₂³×C₆₇	1	4	yes	—
382	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
383	#103	C₂³×C₆₇	1	8	yes	—
384	#103	C₂³×C₆₇	1	8	yes	—
385	#102	C₂³×C₆₇	1	8	yes	—
386	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
387	#97	C₂³×C₆₇	1	4	yes	—
388	#41	—	2	4	no	—
389	#41	—	2	4	no	—
390	#41	—	2	16	no	—
391	#41	—	2	8	no	—
392	#41	—	2	8	no	—
393	#41	—	4	8	no	—
394	#41	—	4	32	no	—
395	#41	—	4	16	no	—
396	C₂×Q₈	—	2	2	no	—
397	C₂×Q₈	—	2	8	no	—
398	C₂×Q₈	—	2	4	no	—
399	C₂×Q₈	—	2	16	no	—
400	C₂×Q₈	—	4	8	no	—
401	C₂×Q₈	—	4	16	no	—
402	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
403	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
404	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
405	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
406	C₂×D₄	—	2	4	no	—
407	C₂×D₄	—	2	8	no	—
408	C₂×D₄	—	2	8	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
412	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
413	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
414	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	8	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	2	8	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
421	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
422	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
423	Q₁₆	—	2	16	no	—
424	Q₁₆	—	2	4	no	—
425	Q₁₆	—	4	2	no	—
426	Q₁₆	C₂×D₄×C₆₇	1	16	yes	—
427	SD₁₆	—	2	16	no	—
428	SD₁₆	—	2	8	no	—
429	SD₁₆	—	4	4	no	—
430	D₈	—	2	4	no	—
431	D₈	—	4	2	no	—
432	M₁₆	—	2	4	no	—
433	M₁₆	—	2	8	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	8	no	—
436	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
437	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
438	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
439	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
440	C₄⋊C₄	—	2	4	no	—
441	C₄⋊C₄	—	2	4	no	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	8	no	—
446	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
447	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
448	C₂²⋊C₄	—	2	8	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	8	no	—
452	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
453	C₄²	—	2	4	no	—
454	C₄²	—	2	4	no	—
455	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
456	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
457	C₄²	C₂⁴×C₆₇	1	2	yes	—
458	C₂³	—	2	4	no	—
459	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
460	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
461	Q₈	—	2	8	no	—
462	Q₈	—	2	4	no	—
463	Q₈	—	2	4	no	—
464	Q₈	—	2	4	no	—
465	Q₈	—	2	4	no	—
466	Q₈	—	4	16	no	—
467	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
468	D₄	—	2	8	no	—
469	D₄	—	2	4	no	—
470	D₄	—	2	4	no	—
471	D₄	—	2	4	no	—
472	D₄	—	4	4	no	—
473	D₄	—	4	16	no	—
474	D₄	—	4	4	no	—
475	D₄	—	4	8	no	—
476	D₄	—	8	2	no	—
477	D₄	—	8	8	no	—
478	C₂×C₄	—	2	8	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	16	no	—
486	C₂×C₄	—	4	16	no	—
487	C₂×C₄	—	4	8	no	—
488	C₂×C₄	—	8	2	no	—
489	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
490	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
491	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
492	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
493	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
494	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
495	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
496	C₈	—	2	4	no	—
497	C₈	—	2	2	no	—
498	C₈	C₂²×D₄×C₆₇	1	4	yes	—
499	C₂²	—	2	2	no	—
500	C₂²	—	2	8	no	—
501	C₂²	—	2	4	no	—
502	C₂²	—	4	4	no	—
503	C₂²	—	4	4	no	—
504	C₂²	—	4	4	no	—
505	C₂²	D₄²×C₆₇	1	2	yes	—
506	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
507	C₄	—	2	2	no	—
508	C₄	—	2	2	no	—
509	C₄	—	2	8	no	—
510	C₄	—	2	4	no	—
511	C₄	—	4	4	no	—
512	C₄	—	4	4	no	—
513	C₄	D₄²×C₆₇	1	2	yes	—
514	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
515	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
516	C₂	—	2	2	no	—
517	C₂	—	4	4	no	—
518	C₂	—	8	4	no	—
519	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
520	C₂	C₆₇×#3586	1	2	yes	—

	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×#3271	C₂	semidirect
4. C₆₇×#3270	C₂	semidirect
5. C₆₇×#3150	C₂	semidirect
6. C₆₇×#3060	C₂	semidirect
7. C₆₇×#3037	C₂	semidirect
8. C₆₇×#3013	C₂	semidirect
9. C₂×#441×C₆₇	C₂	semidirect
10. C₆₇×#2932	C₂	semidirect
69. C₂×Q₁₆×C₆₇	D₄	semidirect
255. #61081	C₆₇	direct (N × H)
261. C₂×#578	C₂×C₆₇	semidirect
262. #3497	C₂×C₆₇	semidirect
263. #3271	C₂×C₆₇	semidirect
264. #3270	C₂×C₆₇	semidirect
265. #3150	C₂×C₆₇	semidirect
266. #3060	C₂×C₆₇	semidirect
267. #3037	C₂×C₆₇	semidirect
268. #3013	C₂×C₆₇	semidirect
269. C₂×#441	C₂×C₆₇	semidirect
270. #2932	C₂×C₆₇	semidirect
271. C₆₇	#61081	direct (N × H)
330. C₂×Q₁₆	D₄×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×#3271	2
C₆₇×#3270	2
C₆₇×#3150	2
C₆₇×#3060	2
C₆₇×#3037	2
C₆₇×#3013	2
C₂×#441×C₆₇	2
C₆₇×#2932	2
#61081	67

Subgroups — C67×#61081 #30740332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61081 #30740332