Subgroups — C₆₇×#34188 #30713439

Order 17152.

2542 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2355	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2354	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2352	C₂	1	8	yes	maximal, complemented
5	C₆₇×#2314	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2311	C₂	1	2	yes	maximal
7	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2285	C₂	1	4	yes	maximal, complemented
9	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2274	C₂	1	2	yes	maximal, complemented
11	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
12	C₄×#41×C₆₇	C₄	1	16	yes	complemented
13	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
15	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
16	C₂×#117×C₆₇	C₂²	1	2	yes	—
17	C₂×#117×C₆₇	C₂²	1	8	yes	—
18	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#117×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
22	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
24	#391×C₆₇	C₂²	1	4	yes	—
25	#390×C₆₇	C₂²	1	4	yes	—
26	#388×C₆₇	C₂²	1	16	yes	—
27	#387×C₆₇	C₂²	1	16	yes	—
28	#386×C₆₇	C₂²	1	8	yes	—
29	#384×C₆₇	C₂²	1	2	yes	—
30	#384×C₆₇	C₂²	1	2	yes	—
31	#384×C₆₇	C₂²	1	4	yes	—
32	#384×C₆₇	C₂²	1	2	yes	—
33	#382×C₆₇	C₂²	1	2	yes	—
34	#382×C₆₇	C₂²	1	8	yes	—
35	#380×C₆₇	C₂²	1	8	yes	—
36	#380×C₆₇	C₂²	1	4	yes	—
37	#380×C₆₇	C₂²	1	8	yes	—
38	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	—
39	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	—
40	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	—
41	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
42	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
43	C₂×#41×C₆₇	—	2	4	no	—
44	C₂×#41×C₆₇	—	2	4	no	—
45	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
46	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
48	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
51	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
52	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₄×Q₈×C₆₇	C₂×C₄	1	16	yes	complemented
55	C₄×D₄×C₆₇	C₂×C₄	1	32	yes	—
56	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
57	#117×C₆₇	—	2	8	no	—
58	#117×C₆₇	—	4	16	no	—
59	#117×C₆₇	C₂×C₄	1	8	yes	—
60	#117×C₆₇	C₂×C₄	1	16	yes	complemented
61	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
62	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
72	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
73	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
74	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
75	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
76	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	—
77	C₂×C₄²×C₆₇	C₂×C₄	1	16	yes	complemented
78	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
79	C₂×C₄²×C₆₇	C₂³	1	16	yes	—
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
82	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
84	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
85	#95×C₆₇	C₂³	1	16	yes	—
86	#95×C₆₇	C₂³	1	16	yes	—
87	#95×C₆₇	C₂³	1	8	yes	—
88	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
89	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
90	#41×C₆₇	—	2	32	no	—
91	C₂×Q₈×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	—	2	2	no	—
93	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
94	C₂×D₄×C₆₇	—	2	4	no	—
95	C₂×D₄×C₆₇	—	2	4	no	—
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
99	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
100	C₂²×C₄×C₆₇	—	2	4	no	—
101	C₂²×C₄×C₆₇	—	2	4	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	8	no	—
104	C₂²×C₄×C₆₇	—	2	4	no	—
105	C₂²×C₄×C₆₇	—	2	8	no	—
106	C₂²×C₄×C₆₇	—	2	8	no	—
107	C₂²×C₄×C₆₇	—	4	8	no	—
108	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
109	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
110	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	—
111	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
112	C₂²×C₄×C₆₇	#41	1	4	yes	—
113	C₂²×C₄×C₆₇	#41	1	4	yes	—
114	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
115	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
116	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
117	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
118	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
122	C₄⋊C₄×C₆₇	—	2	8	no	—
123	C₄⋊C₄×C₆₇	—	2	8	no	—
124	C₄⋊C₄×C₆₇	—	2	8	no	—
125	C₄⋊C₄×C₆₇	—	2	16	no	—
126	C₄⋊C₄×C₆₇	—	2	8	no	—
127	C₄⋊C₄×C₆₇	—	2	8	no	—
128	C₄⋊C₄×C₆₇	—	4	8	no	—
129	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
130	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
131	C₂²⋊C₄×C₆₇	—	2	8	no	—
132	C₂²⋊C₄×C₆₇	—	4	16	no	—
133	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
134	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
135	C₄²×C₆₇	—	2	8	no	—
136	C₄²×C₆₇	—	4	16	no	—
137	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
138	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
139	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
140	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
141	C₄²×C₆₇	C₂²×C₄	1	16	yes	—
142	C₂³×C₆₇	—	2	2	no	—
143	C₂³×C₆₇	—	2	2	no	—
144	C₂³×C₆₇	—	2	2	no	—
145	C₂³×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	—	2	4	no	—
147	C₂³×C₆₇	—	2	4	no	—
148	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
149	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
150	C₂³×C₆₇	C₂×#41	1	2	yes	—
151	C₂³×C₆₇	C₂×#41	1	2	yes	—
152	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
153	Q₈×C₆₇	—	2	8	no	—
154	D₄×C₆₇	—	2	16	no	—
155	D₄×C₆₇	—	2	8	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	4	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	4	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	16	no	—
163	C₂×C₄×C₆₇	—	2	16	no	—
164	C₂×C₄×C₆₇	—	2	32	no	—
165	C₂×C₄×C₆₇	—	2	8	no	—
166	C₂×C₄×C₆₇	—	2	8	no	—
167	C₂×C₄×C₆₇	—	2	8	no	—
168	C₂×C₄×C₆₇	—	2	16	no	—
169	C₂×C₄×C₆₇	—	2	8	no	—
170	C₂×C₄×C₆₇	—	8	8	no	—
171	C₂×C₄×C₆₇	#117	1	16	yes	complemented
172	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
173	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
174	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
175	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
176	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
177	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
178	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
179	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
180	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
181	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
182	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
183	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
184	C₂²×C₆₇	—	2	4	no	—
185	C₂²×C₆₇	—	2	2	no	—
186	C₂²×C₆₇	—	2	2	no	—
187	C₂²×C₆₇	—	2	4	no	—
188	C₂²×C₆₇	—	2	4	no	—
189	C₂²×C₆₇	—	2	8	no	—
190	C₂²×C₆₇	—	4	2	no	—
191	C₂²×C₆₇	—	4	2	no	—
192	C₂²×C₆₇	—	4	4	no	—
193	C₂²×C₆₇	C₂×#117	1	4	yes	—
194	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
195	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
196	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
197	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
198	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
199	C₄×C₆₇	—	2	4	no	—
200	C₄×C₆₇	—	2	8	no	—
201	C₄×C₆₇	—	2	4	no	—
202	C₄×C₆₇	—	4	8	no	—
203	C₄×C₆₇	—	4	8	no	—
204	C₄×C₆₇	C₂×#117	1	4	yes	—
205	C₄×C₆₇	C₂×#117	1	4	yes	—
206	C₄×C₆₇	#520	1	8	yes	—
207	#34188	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂×C₆₇	—	2	4	no	—
209	C₂×C₆₇	—	4	4	no	—
210	C₂×C₆₇	C₂²×#117	1	1	yes	characteristic
211	C₂×C₆₇	C₂×#520	1	2	yes	—
212	C₂×C₆₇	#3508	1	1	yes	characteristic
213	C₂×C₆₇	#3509	1	1	yes	characteristic
214	C₂×C₆₇	#3511	1	2	yes	—
215	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic
216	#2355	C₂×C₆₇	1	4	yes	complemented
217	#2354	C₂×C₆₇	1	4	yes	complemented
218	#2352	C₂×C₆₇	1	8	yes	complemented
219	#2314	C₂×C₆₇	1	4	yes	complemented
220	#2311	C₂×C₆₇	1	2	yes	—
221	#2285	C₂×C₆₇	1	1	yes	characteristic, complemented
222	#2285	C₂×C₆₇	1	4	yes	complemented
223	C₂×#384	C₂×C₆₇	1	1	yes	characteristic, complemented
224	#2274	C₂×C₆₇	1	2	yes	complemented
225	C₆₇	#34188	1	1	yes	characteristic, direct product factor, Sylow
226	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
227	C₄×#41	C₄×C₆₇	1	16	yes	complemented
228	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
229	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
230	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
231	C₂×#117	C₂²×C₆₇	1	2	yes	—
232	C₂×#117	C₂²×C₆₇	1	8	yes	—
233	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#117	C₂²×C₆₇	1	2	yes	—
235	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
236	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
237	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
238	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
239	#391	C₂²×C₆₇	1	4	yes	—
240	#390	C₂²×C₆₇	1	4	yes	—
241	#388	C₂²×C₆₇	1	16	yes	—
242	#387	C₂²×C₆₇	1	16	yes	—
243	#386	C₂²×C₆₇	1	8	yes	—
244	#384	C₂²×C₆₇	1	2	yes	—
245	#384	C₂²×C₆₇	1	4	yes	—
246	#384	C₂²×C₆₇	1	2	yes	—
247	#384	C₂²×C₆₇	1	2	yes	—
248	#382	C₂²×C₆₇	1	2	yes	—
249	#382	C₂²×C₆₇	1	8	yes	—
250	#380	C₂²×C₆₇	1	8	yes	—
251	#380	C₂²×C₆₇	1	4	yes	—
252	#380	C₂²×C₆₇	1	8	yes	—
253	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
254	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
255	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
256	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
257	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
258	C₂×#41	—	2	4	no	—
259	C₂×#41	—	2	4	no	—
260	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
261	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
262	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
263	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
264	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
265	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
266	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
267	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
268	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
269	C₄×Q₈	C₂×C₄×C₆₇	1	16	yes	complemented
270	C₄×D₄	C₂×C₄×C₆₇	1	32	yes	—
271	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
272	#117	—	2	8	no	—
273	#117	—	4	16	no	—
274	#117	C₂×C₄×C₆₇	1	8	yes	—
275	#117	C₂×C₄×C₆₇	1	16	yes	complemented
276	C₂×C₄⋊C₄	—	2	8	no	—
277	C₂×C₄⋊C₄	—	4	8	no	—
278	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
279	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
280	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
281	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
282	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
283	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
284	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
285	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
286	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
287	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
288	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
289	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
290	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
291	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	—
292	C₂×C₄²	C₂×C₄×C₆₇	1	16	yes	complemented
293	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
294	C₂×C₄²	C₂³×C₆₇	1	16	yes	—
295	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
296	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
297	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
298	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
299	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
300	#95	C₂³×C₆₇	1	16	yes	—
301	#95	C₂³×C₆₇	1	16	yes	—
302	#95	C₂³×C₆₇	1	8	yes	—
303	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
304	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
305	#41	—	2	32	no	—
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	—	2	2	no	—
308	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
309	C₂×D₄	—	2	4	no	—
310	C₂×D₄	—	2	4	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
314	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
315	C₂²×C₄	—	2	4	no	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	2	4	no	—
318	C₂²×C₄	—	2	8	no	—
319	C₂²×C₄	—	2	4	no	—
320	C₂²×C₄	—	2	8	no	—
321	C₂²×C₄	—	2	8	no	—
322	C₂²×C₄	—	4	8	no	—
323	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
324	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
325	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	—
326	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
327	C₂²×C₄	#41×C₆₇	1	4	yes	—
328	C₂²×C₄	#41×C₆₇	1	4	yes	—
329	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
330	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
331	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
332	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
333	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
334	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
335	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
336	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
337	C₄⋊C₄	—	2	8	no	—
338	C₄⋊C₄	—	2	8	no	—
339	C₄⋊C₄	—	2	8	no	—
340	C₄⋊C₄	—	2	16	no	—
341	C₄⋊C₄	—	2	8	no	—
342	C₄⋊C₄	—	2	8	no	—
343	C₄⋊C₄	—	4	8	no	—
344	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
345	C₄⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
346	C₂²⋊C₄	—	2	8	no	—
347	C₂²⋊C₄	—	4	16	no	—
348	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
349	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
350	C₄²	—	2	8	no	—
351	C₄²	—	4	16	no	—
352	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
353	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
354	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
355	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
356	C₄²	C₂²×C₄×C₆₇	1	16	yes	—
357	C₂³	—	2	2	no	—
358	C₂³	—	2	2	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	2	2	no	—
361	C₂³	—	2	4	no	—
362	C₂³	—	2	4	no	—
363	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
364	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
365	C₂³	C₂×#41×C₆₇	1	2	yes	—
366	C₂³	C₂×#41×C₆₇	1	2	yes	—
367	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
368	Q₈	—	2	8	no	—
369	D₄	—	2	16	no	—
370	D₄	—	2	8	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	16	no	—
378	C₂×C₄	—	2	16	no	—
379	C₂×C₄	—	2	32	no	—
380	C₂×C₄	—	2	8	no	—
381	C₂×C₄	—	2	8	no	—
382	C₂×C₄	—	2	8	no	—
383	C₂×C₄	—	2	16	no	—
384	C₂×C₄	—	2	8	no	—
385	C₂×C₄	—	8	8	no	—
386	C₂×C₄	#117×C₆₇	1	16	yes	complemented
387	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
388	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
389	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
390	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
391	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
392	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
393	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
394	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
395	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
396	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
397	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
398	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
399	C₂²	—	2	4	no	—
400	C₂²	—	2	2	no	—
401	C₂²	—	2	2	no	—
402	C₂²	—	2	4	no	—
403	C₂²	—	2	4	no	—
404	C₂²	—	2	8	no	—
405	C₂²	—	4	2	no	—
406	C₂²	—	4	2	no	—
407	C₂²	—	4	4	no	—
408	C₂²	C₂×#117×C₆₇	1	4	yes	—
409	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
410	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
411	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
412	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
413	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
414	C₄	—	2	4	no	—
415	C₄	—	2	8	no	—
416	C₄	—	2	4	no	—
417	C₄	—	4	8	no	—
418	C₄	—	4	8	no	—
419	C₄	C₂×#117×C₆₇	1	4	yes	—
420	C₄	C₂×#117×C₆₇	1	4	yes	—
421	C₄	#520×C₆₇	1	8	yes	—
422	C₂	—	2	4	no	—
423	C₂	—	4	4	no	—
424	C₂	C₂²×#117×C₆₇	1	1	yes	characteristic
425	C₂	C₂×#520×C₆₇	1	2	yes	—
426	C₂	C₆₇×#3508	1	1	yes	characteristic
427	C₂	C₆₇×#3509	1	1	yes	characteristic
428	C₂	C₆₇×#3511	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2355	C₂	semidirect
3. C₆₇×#2354	C₂	semidirect
4. C₆₇×#2352	C₂	semidirect
5. C₆₇×#2314	C₂	semidirect
7. C₆₇×#2285	C₂	semidirect
8. C₆₇×#2285	C₂	semidirect
9. C₂×#384×C₆₇	C₂	semidirect
10. C₆₇×#2274	C₂	semidirect
12. C₄×#41×C₆₇	C₄	semidirect
54. C₄×Q₈×C₆₇	C₂×C₄	semidirect
56. C₄×D₄×C₆₇	C₂×C₄	semidirect
60. #117×C₆₇	C₂×C₄	semidirect
77. C₂×C₄²×C₆₇	C₂×C₄	semidirect
171. C₂×C₄×C₆₇	#117	semidirect
207. #34188	C₆₇	direct (N × H)
216. #2355	C₂×C₆₇	semidirect
217. #2354	C₂×C₆₇	semidirect
218. #2352	C₂×C₆₇	semidirect
219. #2314	C₂×C₆₇	semidirect
221. #2285	C₂×C₆₇	semidirect
222. #2285	C₂×C₆₇	semidirect
223. C₂×#384	C₂×C₆₇	semidirect
224. #2274	C₂×C₆₇	semidirect
225. C₆₇	#34188	direct (N × H)
227. C₄×#41	C₄×C₆₇	semidirect
269. C₄×Q₈	C₂×C₄×C₆₇	semidirect
271. C₄×D₄	C₂×C₄×C₆₇	semidirect
275. #117	C₂×C₄×C₆₇	semidirect
292. C₂×C₄²	C₂×C₄×C₆₇	semidirect
386. C₂×C₄	#117×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₆₇	2
C₆₇×#2355	2
C₆₇×#2354	2
C₆₇×#2352	2
C₆₇×#2314	2
C₆₇×#2311	2
C₆₇×#2285	2
C₂×#384×C₆₇	2
C₆₇×#2274	2
#34188	67

Subgroups — C67×#34188 #30713439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34188 #30713439