Subgroups — C₆₇×#59821 #30739072

Order 17152.

2244 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#556×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2727	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2548	C₂	1	2	yes	maximal
4	C₆₇×#2542	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2538	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2512	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2448	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2415	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2354	C₂	1	2	yes	maximal, complemented
10	Q₈×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#127×C₆₇	C₂	1	1	yes	maximal, characteristic
14	#556×C₆₇	—	2	16	no	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#126×C₆₇	C₂²	1	8	yes	—
19	C₂×#124×C₆₇	C₂²	1	4	yes	—
20	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
22	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
28	C₂×#117×C₆₇	C₂²	1	4	yes	—
29	#399×C₆₇	C₂²	1	8	yes	—
30	#397×C₆₇	C₂²	1	8	yes	—
31	#396×C₆₇	C₂²	1	8	yes	—
32	#394×C₆₇	C₂²	1	8	yes	—
33	#391×C₆₇	C₂²	1	2	yes	—
34	#391×C₆₇	C₂²	1	2	yes	—
35	#390×C₆₇	C₂²	1	2	yes	—
36	#390×C₆₇	C₂²	1	4	yes	—
37	#390×C₆₇	C₂²	1	4	yes	—
38	#390×C₆₇	C₂²	1	4	yes	—
39	#390×C₆₇	C₂²	1	2	yes	—
40	#390×C₆₇	C₂²	1	2	yes	—
41	#389×C₆₇	C₂²	1	2	yes	—
42	#389×C₆₇	C₂²	1	4	yes	—
43	#389×C₆₇	C₂²	1	2	yes	—
44	#389×C₆₇	C₂²	1	2	yes	—
45	#389×C₆₇	C₂²	1	2	yes	—
46	#389×C₆₇	C₂²	1	8	yes	—
47	#388×C₆₇	C₂²	1	8	yes	—
48	#388×C₆₇	C₂²	1	8	yes	—
49	#384×C₆₇	C₂²	1	4	yes	—
50	#384×C₆₇	C₂²	1	1	yes	characteristic
51	#384×C₆₇	C₂²	1	1	yes	characteristic
52	#384×C₆₇	C₂²	1	1	yes	characteristic
53	#380×C₆₇	C₂²	1	4	yes	—
54	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
56	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
57	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
58	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
59	C₄³×C₆₇	C₂²	1	1	yes	characteristic
60	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
61	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
62	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
63	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
64	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
65	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
66	C₄⋊Q₈×C₆₇	—	2	4	no	—
67	C₄⋊Q₈×C₆₇	D₄	1	8	yes	complemented
68	#127×C₆₇	—	2	4	no	—
69	#126×C₆₇	—	2	32	no	—
70	#124×C₆₇	—	2	16	no	—
71	C₂²⋊Q₈×C₆₇	—	2	16	no	—
72	C₂²⋊Q₈×C₆₇	—	4	8	no	—
73	C₄×Q₈×C₆₇	—	2	16	no	—
74	C₄×Q₈×C₆₇	—	2	8	no	—
75	C₄×Q₈×C₆₇	—	2	8	no	—
76	C₄×D₄×C₆₇	—	2	8	no	—
77	C₄×D₄×C₆₇	—	2	8	no	—
78	C₄×D₄×C₆₇	—	4	4	no	—
79	#117×C₆₇	—	2	16	no	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
97	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
98	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
101	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
104	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
105	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
107	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
108	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
114	#95×C₆₇	C₂³	1	8	yes	—
115	#95×C₆₇	C₂³	1	4	yes	—
116	#95×C₆₇	C₂³	1	4	yes	—
117	#95×C₆₇	C₂³	1	4	yes	—
118	#95×C₆₇	C₂³	1	4	yes	—
119	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
120	C₂×Q₈×C₆₇	—	2	4	no	—
121	C₂×Q₈×C₆₇	—	2	4	no	—
122	C₂×Q₈×C₆₇	—	2	4	no	—
123	C₂×Q₈×C₆₇	—	2	4	no	—
124	C₂×Q₈×C₆₇	—	2	4	no	—
125	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
126	C₂×D₄×C₆₇	—	2	4	no	—
127	C₂×D₄×C₆₇	—	2	4	no	—
128	C₂×D₄×C₆₇	—	2	4	no	—
129	C₂×D₄×C₆₇	—	2	4	no	—
130	C₂×D₄×C₆₇	—	2	4	no	—
131	C₂×D₄×C₆₇	—	2	4	no	—
132	C₂×D₄×C₆₇	—	2	4	no	—
133	C₂×D₄×C₆₇	—	4	2	no	—
134	C₂×D₄×C₆₇	—	4	2	no	—
135	C₂²×C₄×C₆₇	—	2	8	no	—
136	C₂²×C₄×C₆₇	—	2	8	no	—
137	C₂²×C₄×C₆₇	—	2	8	no	—
138	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
139	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
141	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
142	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
145	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
150	C₄⋊C₄×C₆₇	—	2	4	no	—
151	C₄⋊C₄×C₆₇	—	2	4	no	—
152	C₄⋊C₄×C₆₇	—	2	16	no	—
153	C₄⋊C₄×C₆₇	—	2	8	no	—
154	C₄⋊C₄×C₆₇	—	2	4	no	—
155	C₄⋊C₄×C₆₇	—	2	8	no	—
156	C₄⋊C₄×C₆₇	—	2	4	no	—
157	C₄⋊C₄×C₆₇	—	2	4	no	—
158	C₄⋊C₄×C₆₇	—	2	4	no	—
159	C₄⋊C₄×C₆₇	—	2	4	no	—
160	C₄⋊C₄×C₆₇	—	2	4	no	—
161	C₄⋊C₄×C₆₇	—	4	2	no	—
162	C₄⋊C₄×C₆₇	—	4	2	no	—
163	C₄⋊C₄×C₆₇	—	4	2	no	—
164	C₄⋊C₄×C₆₇	—	4	8	no	—
165	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
166	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
167	C₂²⋊C₄×C₆₇	—	2	16	no	—
168	C₂²⋊C₄×C₆₇	—	2	8	no	—
169	C₂²⋊C₄×C₆₇	—	2	16	no	—
170	C₂²⋊C₄×C₆₇	—	2	8	no	—
171	C₂²⋊C₄×C₆₇	—	4	8	no	—
172	C₂²⋊C₄×C₆₇	—	4	4	no	—
173	C₂²⋊C₄×C₆₇	—	4	8	no	—
174	C₄²×C₆₇	—	2	16	no	—
175	C₄²×C₆₇	—	2	8	no	—
176	C₄²×C₆₇	—	2	8	no	—
177	C₄²×C₆₇	—	2	2	no	—
178	C₄²×C₆₇	—	2	2	no	—
179	C₄²×C₆₇	—	2	2	no	—
180	C₄²×C₆₇	—	4	1	no	—
181	C₄²×C₆₇	C₂×D₄	1	4	yes	—
182	C₄²×C₆₇	#41	1	4	yes	—
183	C₄²×C₆₇	#41	1	4	yes	—
184	C₂³×C₆₇	—	2	4	no	—
185	C₂³×C₆₇	—	2	4	no	—
186	C₂³×C₆₇	—	2	4	no	—
187	C₂³×C₆₇	—	2	4	no	—
188	C₂³×C₆₇	—	2	4	no	—
189	C₂³×C₆₇	—	2	4	no	—
190	C₂³×C₆₇	—	2	4	no	—
191	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
192	Q₈×C₆₇	—	2	16	no	—
193	Q₈×C₆₇	—	4	8	no	—
194	D₄×C₆₇	—	4	8	no	—
195	D₄×C₆₇	—	4	8	no	—
196	D₄×C₆₇	—	4	8	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	1	no	—
199	C₂×C₄×C₆₇	—	2	1	no	—
200	C₂×C₄×C₆₇	—	2	1	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	1	no	—
203	C₂×C₄×C₆₇	—	2	1	no	—
204	C₂×C₄×C₆₇	—	2	1	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	2	4	no	—
207	C₂×C₄×C₆₇	—	2	4	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	4	no	—
210	C₂×C₄×C₆₇	—	2	4	no	—
211	C₂×C₄×C₆₇	—	2	1	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	1	no	—
216	C₂×C₄×C₆₇	—	2	1	no	—
217	C₂×C₄×C₆₇	—	2	4	no	—
218	C₂×C₄×C₆₇	—	2	4	no	—
219	C₂×C₄×C₆₇	—	2	4	no	—
220	C₂×C₄×C₆₇	—	4	16	no	—
221	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
222	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
223	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
224	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
226	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
227	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
228	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
229	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
230	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
231	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
232	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
233	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
234	C₂²×C₆₇	—	4	4	no	—
235	C₂²×C₆₇	—	4	4	no	—
236	C₂²×C₆₇	—	4	4	no	—
237	C₂²×C₆₇	—	4	4	no	—
238	C₂²×C₆₇	—	4	4	no	—
239	C₂²×C₆₇	—	4	4	no	—
240	C₂²×C₆₇	—	4	4	no	—
241	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
242	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
243	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
244	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
245	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
246	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
247	C₂²×C₆₇	#585	1	1	yes	characteristic
248	C₄×C₆₇	—	2	8	no	—
249	C₄×C₆₇	—	2	2	no	—
250	C₄×C₆₇	—	2	2	no	—
251	C₄×C₆₇	—	2	2	no	—
252	C₄×C₆₇	—	4	4	no	—
253	C₄×C₆₇	—	4	4	no	—
254	C₄×C₆₇	—	4	4	no	—
255	C₄×C₆₇	—	4	4	no	—
256	C₄×C₆₇	—	4	1	no	—
257	C₄×C₆₇	#547	1	4	yes	—
258	C₄×C₆₇	#550	1	4	yes	—
259	C₄×C₆₇	#556	1	4	yes	complemented
260	#59821	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
261	C₂×C₆₇	—	8	4	no	—
262	C₂×C₆₇	C₂×#547	1	1	yes	characteristic
263	C₂×C₆₇	C₂×#550	1	1	yes	characteristic
264	C₂×C₆₇	C₂×#556	1	1	yes	characteristic
265	C₂×C₆₇	#3497	1	1	yes	characteristic
266	C₂×C₆₇	#3506	1	1	yes	characteristic
267	C₂×C₆₇	#3507	1	1	yes	characteristic
268	C₂×C₆₇	#3550	1	1	yes	characteristic
269	C₂×#556	C₂×C₆₇	1	2	yes	complemented
270	#2727	C₂×C₆₇	1	4	yes	complemented
271	#2548	C₂×C₆₇	1	2	yes	—
272	#2542	C₂×C₆₇	1	2	yes	complemented
273	#2538	C₂×C₆₇	1	2	yes	complemented
274	#2512	C₂×C₆₇	1	4	yes	complemented
275	#2448	C₂×C₆₇	1	4	yes	complemented
276	#2415	C₂×C₆₇	1	4	yes	complemented
277	#2354	C₂×C₆₇	1	2	yes	complemented
278	Q₈×C₄⋊C₄	C₂×C₆₇	1	2	yes	complemented
279	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
280	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
281	C₄×#127	C₂×C₆₇	1	1	yes	characteristic
282	C₆₇	#59821	1	1	yes	characteristic, direct product factor, Sylow
283	#556	—	2	16	no	—
284	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
285	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
286	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
287	C₂×#126	C₂²×C₆₇	1	8	yes	—
288	C₂×#124	C₂²×C₆₇	1	4	yes	—
289	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
290	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
291	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
292	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
293	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
294	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
295	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
296	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
297	C₂×#117	C₂²×C₆₇	1	4	yes	—
298	#399	C₂²×C₆₇	1	8	yes	—
299	#397	C₂²×C₆₇	1	8	yes	—
300	#396	C₂²×C₆₇	1	8	yes	—
301	#394	C₂²×C₆₇	1	8	yes	—
302	#391	C₂²×C₆₇	1	2	yes	—
303	#391	C₂²×C₆₇	1	2	yes	—
304	#390	C₂²×C₆₇	1	2	yes	—
305	#390	C₂²×C₆₇	1	2	yes	—
306	#390	C₂²×C₆₇	1	2	yes	—
307	#390	C₂²×C₆₇	1	4	yes	—
308	#390	C₂²×C₆₇	1	4	yes	—
309	#390	C₂²×C₆₇	1	4	yes	—
310	#389	C₂²×C₆₇	1	2	yes	—
311	#389	C₂²×C₆₇	1	2	yes	—
312	#389	C₂²×C₆₇	1	2	yes	—
313	#389	C₂²×C₆₇	1	2	yes	—
314	#389	C₂²×C₆₇	1	8	yes	—
315	#389	C₂²×C₆₇	1	4	yes	—
316	#388	C₂²×C₆₇	1	8	yes	—
317	#388	C₂²×C₆₇	1	8	yes	—
318	#384	C₂²×C₆₇	1	1	yes	characteristic
319	#384	C₂²×C₆₇	1	1	yes	characteristic
320	#384	C₂²×C₆₇	1	1	yes	characteristic
321	#384	C₂²×C₆₇	1	4	yes	—
322	#380	C₂²×C₆₇	1	4	yes	—
323	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
324	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
325	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
326	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
327	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
328	C₄³	C₂²×C₆₇	1	1	yes	characteristic
329	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
330	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
331	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
332	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
333	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
334	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
335	C₄⋊Q₈	—	2	4	no	—
336	C₄⋊Q₈	D₄×C₆₇	1	8	yes	complemented
337	#127	—	2	4	no	—
338	#126	—	2	32	no	—
339	#124	—	2	16	no	—
340	C₂²⋊Q₈	—	2	16	no	—
341	C₂²⋊Q₈	—	4	8	no	—
342	C₄×Q₈	—	2	16	no	—
343	C₄×Q₈	—	2	8	no	—
344	C₄×Q₈	—	2	8	no	—
345	C₄×D₄	—	2	8	no	—
346	C₄×D₄	—	2	8	no	—
347	C₄×D₄	—	4	4	no	—
348	#117	—	2	16	no	—
349	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
350	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
351	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
352	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
353	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
354	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
355	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
356	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
357	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
358	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
359	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
360	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
361	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
362	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
363	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
364	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
365	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
366	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
367	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
368	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
369	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
370	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
371	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
372	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
373	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
374	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
375	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
376	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
377	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
378	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
379	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
380	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
381	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
382	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
383	#95	C₂³×C₆₇	1	8	yes	—
384	#95	C₂³×C₆₇	1	4	yes	—
385	#95	C₂³×C₆₇	1	4	yes	—
386	#95	C₂³×C₆₇	1	4	yes	—
387	#95	C₂³×C₆₇	1	4	yes	—
388	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
389	C₂×Q₈	—	2	4	no	—
390	C₂×Q₈	—	2	4	no	—
391	C₂×Q₈	—	2	4	no	—
392	C₂×Q₈	—	2	4	no	—
393	C₂×Q₈	—	2	4	no	—
394	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
395	C₂×D₄	—	2	4	no	—
396	C₂×D₄	—	2	4	no	—
397	C₂×D₄	—	2	4	no	—
398	C₂×D₄	—	2	4	no	—
399	C₂×D₄	—	2	4	no	—
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	4	2	no	—
403	C₂×D₄	—	4	2	no	—
404	C₂²×C₄	—	2	8	no	—
405	C₂²×C₄	—	2	8	no	—
406	C₂²×C₄	—	2	8	no	—
407	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
408	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
409	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
410	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
411	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
412	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
413	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
414	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
415	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
416	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
417	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
418	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
419	C₄⋊C₄	—	2	4	no	—
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	—	2	16	no	—
422	C₄⋊C₄	—	2	8	no	—
423	C₄⋊C₄	—	2	4	no	—
424	C₄⋊C₄	—	2	8	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₄⋊C₄	—	2	4	no	—
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	4	no	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	4	2	no	—
431	C₄⋊C₄	—	4	8	no	—
432	C₄⋊C₄	—	4	2	no	—
433	C₄⋊C₄	—	4	2	no	—
434	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
435	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
436	C₂²⋊C₄	—	2	16	no	—
437	C₂²⋊C₄	—	2	8	no	—
438	C₂²⋊C₄	—	2	16	no	—
439	C₂²⋊C₄	—	2	8	no	—
440	C₂²⋊C₄	—	4	4	no	—
441	C₂²⋊C₄	—	4	8	no	—
442	C₂²⋊C₄	—	4	8	no	—
443	C₄²	—	2	16	no	—
444	C₄²	—	2	8	no	—
445	C₄²	—	2	8	no	—
446	C₄²	—	2	2	no	—
447	C₄²	—	2	2	no	—
448	C₄²	—	2	2	no	—
449	C₄²	—	4	1	no	—
450	C₄²	C₂×D₄×C₆₇	1	4	yes	—
451	C₄²	#41×C₆₇	1	4	yes	—
452	C₄²	#41×C₆₇	1	4	yes	—
453	C₂³	—	2	4	no	—
454	C₂³	—	2	4	no	—
455	C₂³	—	2	4	no	—
456	C₂³	—	2	4	no	—
457	C₂³	—	2	4	no	—
458	C₂³	—	2	4	no	—
459	C₂³	—	2	4	no	—
460	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
461	Q₈	—	2	16	no	—
462	Q₈	—	4	8	no	—
463	D₄	—	4	8	no	—
464	D₄	—	4	8	no	—
465	D₄	—	4	8	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	4	16	no	—
490	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
491	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
492	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
493	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
494	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
495	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
496	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
497	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
498	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
499	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
500	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
501	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
502	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
503	C₂²	—	4	4	no	—
504	C₂²	—	4	4	no	—
505	C₂²	—	4	4	no	—
506	C₂²	—	4	4	no	—
507	C₂²	—	4	4	no	—
508	C₂²	—	4	4	no	—
509	C₂²	—	4	4	no	—
510	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
511	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
512	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
513	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
514	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
515	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
516	C₂²	#585×C₆₇	1	1	yes	characteristic
517	C₄	—	2	8	no	—
518	C₄	—	2	2	no	—
519	C₄	—	2	2	no	—
520	C₄	—	2	2	no	—
521	C₄	—	4	1	no	—
522	C₄	—	4	4	no	—
523	C₄	—	4	4	no	—
524	C₄	—	4	4	no	—
525	C₄	—	4	4	no	—
526	C₄	#547×C₆₇	1	4	yes	—
527	C₄	#550×C₆₇	1	4	yes	—
528	C₄	#556×C₆₇	1	4	yes	complemented
529	C₂	—	8	4	no	—
530	C₂	C₂×#547×C₆₇	1	1	yes	characteristic
531	C₂	C₂×#550×C₆₇	1	1	yes	characteristic
532	C₂	C₂×#556×C₆₇	1	1	yes	characteristic
533	C₂	C₆₇×#3497	1	1	yes	characteristic
534	C₂	C₆₇×#3506	1	1	yes	characteristic
535	C₂	C₆₇×#3507	1	1	yes	characteristic
536	C₂	C₆₇×#3550	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#556×C₆₇	C₂	semidirect
2. C₆₇×#2727	C₂	semidirect
4. C₆₇×#2542	C₂	semidirect
5. C₆₇×#2538	C₂	semidirect
6. C₆₇×#2512	C₂	semidirect
7. C₆₇×#2448	C₂	semidirect
8. C₆₇×#2415	C₂	semidirect
9. C₆₇×#2354	C₂	semidirect
10. Q₈×C₄⋊C₄×C₆₇	C₂	semidirect
11. C₄×C₄⋊Q₈×C₆₇	C₂	semidirect
12. C₄×C₄⋊Q₈×C₆₇	C₂	semidirect
67. C₄⋊Q₈×C₆₇	D₄	semidirect
259. C₄×C₆₇	#556	semidirect
260. #59821	C₆₇	direct (N × H)
269. C₂×#556	C₂×C₆₇	semidirect
270. #2727	C₂×C₆₇	semidirect
272. #2542	C₂×C₆₇	semidirect
273. #2538	C₂×C₆₇	semidirect
274. #2512	C₂×C₆₇	semidirect
275. #2448	C₂×C₆₇	semidirect
276. #2415	C₂×C₆₇	semidirect
277. #2354	C₂×C₆₇	semidirect
278. Q₈×C₄⋊C₄	C₂×C₆₇	semidirect
279. C₄×C₄⋊Q₈	C₂×C₆₇	semidirect
280. C₄×C₄⋊Q₈	C₂×C₆₇	semidirect
282. C₆₇	#59821	direct (N × H)
336. C₄⋊Q₈	D₄×C₆₇	semidirect
528. C₄	#556×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#556×C₆₇	2
C₆₇×#2727	2
C₆₇×#2548	2
C₆₇×#2542	2
C₆₇×#2538	2
C₆₇×#2512	2
C₆₇×#2448	2
C₆₇×#2415	2
C₆₇×#2354	2
Q₈×C₄⋊C₄×C₆₇	2
C₄×C₄⋊Q₈×C₆₇	2
C₄×#127×C₆₇	2
#59821	67

Subgroups — C67×#59821 #30739072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59821 #30739072