Subgroups — C₆₇×#60619 #30739870

Order 17152.

2206 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3458	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#535×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3140	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3138	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3136	C₂	1	16	yes	maximal, complemented
6	C₆₇×#2896	C₂	1	2	yes	maximal, complemented
7	C₂×#418×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2873	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
12	#544×C₆₇	—	2	2	no	—
13	#544×C₆₇	C₂²	1	8	yes	complemented
14	#543×C₆₇	—	2	2	no	—
15	#535×C₆₇	—	2	4	no	—
16	#535×C₆₇	C₂²	1	8	yes	—
17	#533×C₆₇	—	2	4	no	—
18	#533×C₆₇	C₂²	1	4	yes	complemented
19	C₂×#127×C₆₇	C₂²	1	2	yes	—
20	C₂×#124×C₆₇	C₂²	1	2	yes	—
21	C₂×#121×C₆₇	—	2	2	no	—
22	C₂×#120×C₆₇	—	2	2	no	—
23	C₂×#117×C₆₇	C₂²	1	2	yes	—
24	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
25	#490×C₆₇	C₂²	1	16	yes	complemented
26	#489×C₆₇	C₂²	1	16	yes	complemented
27	#488×C₆₇	C₂²	1	16	yes	complemented
28	#486×C₆₇	C₂²	1	16	yes	complemented
29	#419×C₆₇	C₂²	1	4	yes	complemented
30	#418×C₆₇	C₂²	1	8	yes	—
31	#418×C₆₇	C₂²	1	8	yes	complemented
32	#418×C₆₇	C₂²	1	4	yes	complemented
33	#417×C₆₇	C₂²	1	8	yes	complemented
34	#416×C₆₇	C₂²	1	8	yes	complemented
35	C₂×#102×C₆₇	C₂²	1	8	yes	complemented
36	#405×C₆₇	C₂²	1	8	yes	complemented
37	C₄×M₁₆×C₆₇	C₂²	1	4	yes	complemented
38	C₂⁵×C₆₇	—	2	1	no	—
39	C₂×#41×C₆₇	—	2	2	no	—
40	C₂×#41×C₆₇	—	2	8	no	—
41	C₂×#41×C₆₇	C₂³	1	4	yes	—
42	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₆₇	—	2	1	no	—
44	C₂²×D₄×C₆₇	—	2	1	no	—
45	C₂²×D₄×C₆₇	—	2	2	no	—
46	C₂²×D₄×C₆₇	—	2	4	no	—
47	C₂²×D₄×C₆₇	—	2	8	no	—
48	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
49	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
51	C₂³×C₄×C₆₇	—	2	1	no	—
52	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₆₇	—	2	8	no	—
54	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
55	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
56	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
57	C₄⋊Q₈×C₆₇	—	2	8	no	—
58	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	complemented
59	#127×C₆₇	—	2	4	no	—
60	#127×C₆₇	C₂³	1	8	yes	—
61	#125×C₆₇	—	2	8	no	—
62	#125×C₆₇	C₂³	1	8	yes	complemented
63	#124×C₆₇	—	2	4	no	—
64	#124×C₆₇	C₂³	1	8	yes	—
65	C₂²⋊Q₈×C₆₇	—	2	8	no	—
66	C₂²⋊Q₈×C₆₇	—	2	16	no	—
67	#121×C₆₇	—	2	16	no	—
68	#120×C₆₇	—	2	16	no	—
69	C₄×Q₈×C₆₇	—	2	8	no	—
70	#117×C₆₇	—	2	2	no	—
71	#117×C₆₇	—	2	2	no	—
72	#117×C₆₇	D₄	1	4	yes	—
73	#117×C₆₇	C₂³	1	4	yes	—
74	#117×C₆₇	C₂³	1	4	yes	—
75	#117×C₆₇	C₂³	1	4	yes	—
76	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
78	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
80	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
81	C₂×C₄²×C₆₇	—	2	2	no	—
82	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
83	#103×C₆₇	C₂³	1	16	yes	complemented
84	#102×C₆₇	C₂³	1	32	yes	—
85	#102×C₆₇	C₂³	1	16	yes	complemented
86	#97×C₆₇	C₂³	1	8	yes	complemented
87	C₄×C₈×C₆₇	C₂³	1	8	yes	complemented
88	C₂⁴×C₆₇	—	2	2	no	—
89	C₂⁴×C₆₇	—	2	1	no	—
90	C₂⁴×C₆₇	—	2	4	no	—
91	C₂⁴×C₆₇	—	4	4	no	—
92	C₂⁴×C₆₇	—	8	4	no	—
93	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
94	#41×C₆₇	—	2	8	no	—
95	#41×C₆₇	—	2	8	no	—
96	#41×C₆₇	—	4	8	no	—
97	C₂×Q₈×C₆₇	—	2	1	no	—
98	C₂×Q₈×C₆₇	—	2	8	no	—
99	C₂×Q₈×C₆₇	—	2	4	no	—
100	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
101	C₂×D₄×C₆₇	—	2	4	no	—
102	C₂×D₄×C₆₇	—	2	4	no	—
103	C₂×D₄×C₆₇	—	2	2	no	—
104	C₂×D₄×C₆₇	—	2	1	no	—
105	C₂×D₄×C₆₇	—	2	16	no	—
106	C₂×D₄×C₆₇	—	2	8	no	—
107	C₂×D₄×C₆₇	—	2	8	no	—
108	C₂×D₄×C₆₇	—	2	8	no	—
109	C₂×D₄×C₆₇	—	2	4	no	—
110	C₂×D₄×C₆₇	—	4	4	no	—
111	C₂×D₄×C₆₇	—	4	4	no	—
112	C₂×D₄×C₆₇	—	4	8	no	—
113	C₂×D₄×C₆₇	—	8	8	no	—
114	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
115	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
116	C₂²×C₄×C₆₇	—	2	1	no	—
117	C₂²×C₄×C₆₇	—	2	4	no	—
118	C₂²×C₄×C₆₇	—	2	4	no	—
119	C₂²×C₄×C₆₇	—	4	4	no	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	#41	1	8	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
126	M₁₆×C₆₇	—	2	8	no	—
127	M₁₆×C₆₇	—	2	8	no	—
128	C₂×C₈×C₆₇	—	2	8	no	—
129	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
130	C₄⋊C₄×C₆₇	—	2	4	no	—
131	C₄⋊C₄×C₆₇	—	2	8	no	—
132	C₄⋊C₄×C₆₇	—	2	16	no	—
133	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
134	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
135	C₂²⋊C₄×C₆₇	—	2	4	no	—
136	C₂²⋊C₄×C₆₇	—	2	4	no	—
137	C₂²⋊C₄×C₆₇	—	2	16	no	—
138	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
139	C₄²×C₆₇	—	2	4	no	—
140	C₄²×C₆₇	—	2	4	no	—
141	C₄²×C₆₇	C₂×D₄	1	4	yes	—
142	C₄²×C₆₇	C₂⁴	1	4	yes	—
143	C₂³×C₆₇	—	2	4	no	—
144	C₂³×C₆₇	—	2	2	no	—
145	C₂³×C₆₇	—	4	2	no	—
146	C₂³×C₆₇	—	4	4	no	—
147	C₂³×C₆₇	—	4	8	no	—
148	C₂³×C₆₇	—	4	2	no	—
149	C₂³×C₆₇	—	4	4	no	—
150	C₂³×C₆₇	—	4	8	no	—
151	C₂³×C₆₇	—	8	8	no	—
152	C₂³×C₆₇	—	8	16	no	—
153	C₂³×C₆₇	#124	1	4	yes	—
154	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
155	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
156	Q₈×C₆₇	—	2	4	no	—
157	Q₈×C₆₇	—	4	2	no	—
158	Q₈×C₆₇	—	8	8	no	—
159	D₄×C₆₇	—	2	8	no	—
160	D₄×C₆₇	—	2	4	no	—
161	D₄×C₆₇	—	4	8	no	—
162	D₄×C₆₇	—	4	8	no	—
163	D₄×C₆₇	—	4	4	no	—
164	D₄×C₆₇	—	4	2	no	—
165	D₄×C₆₇	—	8	16	no	—
166	C₂×C₄×C₆₇	—	2	8	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	4	4	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	8	no	—
171	C₂×C₄×C₆₇	—	8	4	no	—
172	C₂×C₄×C₆₇	#124	1	4	yes	—
173	C₂×C₄×C₆₇	#124	1	8	yes	—
174	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
175	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
177	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
178	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
179	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
180	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
181	C₈×C₆₇	—	2	8	no	—
182	C₂²×C₆₇	—	2	2	no	—
183	C₂²×C₆₇	—	2	4	no	—
184	C₂²×C₆₇	—	4	4	no	—
185	C₂²×C₆₇	—	4	4	no	—
186	C₂²×C₆₇	—	4	8	no	—
187	C₂²×C₆₇	—	4	2	no	—
188	C₂²×C₆₇	—	4	4	no	—
189	C₂²×C₆₇	—	8	4	no	—
190	C₂²×C₆₇	—	8	8	no	—
191	C₂²×C₆₇	—	8	16	no	—
192	C₂²×C₆₇	C₂×#124	1	2	yes	—
193	C₂²×C₆₇	C₂×#124	1	4	yes	—
194	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
195	C₄×C₆₇	—	2	8	no	—
196	C₄×C₆₇	—	4	8	no	—
197	C₄×C₆₇	—	4	2	no	—
198	C₄×C₆₇	C₂×#124	1	4	yes	—
199	C₄×C₆₇	C₂×#124	1	2	yes	—
200	C₄×C₆₇	C₂²×#41	1	2	yes	—
201	#60619	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂×C₆₇	—	2	2	no	—
203	C₂×C₆₇	—	2	4	no	—
204	C₂×C₆₇	—	8	2	no	—
205	C₂×C₆₇	—	8	4	no	—
206	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
207	C₂×C₆₇	#3585	1	2	yes	—
208	#3458	C₂×C₆₇	1	1	yes	characteristic, complemented
209	C₂×#535	C₂×C₆₇	1	1	yes	characteristic
210	#3140	C₂×C₆₇	1	4	yes	complemented
211	#3138	C₂×C₆₇	1	4	yes	complemented
212	#3136	C₂×C₆₇	1	16	yes	complemented
213	#2896	C₂×C₆₇	1	2	yes	complemented
214	C₂×#418	C₂×C₆₇	1	2	yes	complemented
215	#2873	C₂×C₆₇	1	1	yes	characteristic, complemented
216	C₆₇	#60619	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
218	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
219	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
220	#544	—	2	2	no	—
221	#544	C₂²×C₆₇	1	8	yes	complemented
222	#543	—	2	2	no	—
223	#535	—	2	4	no	—
224	#535	C₂²×C₆₇	1	8	yes	—
225	#533	—	2	4	no	—
226	#533	C₂²×C₆₇	1	4	yes	complemented
227	C₂×#127	C₂²×C₆₇	1	2	yes	—
228	C₂×#124	C₂²×C₆₇	1	2	yes	—
229	C₂×#121	—	2	2	no	—
230	C₂×#120	—	2	2	no	—
231	C₂×#117	C₂²×C₆₇	1	2	yes	—
232	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
233	#490	C₂²×C₆₇	1	16	yes	complemented
234	#489	C₂²×C₆₇	1	16	yes	complemented
235	#488	C₂²×C₆₇	1	16	yes	complemented
236	#486	C₂²×C₆₇	1	16	yes	complemented
237	#419	C₂²×C₆₇	1	4	yes	complemented
238	#418	C₂²×C₆₇	1	8	yes	—
239	#418	C₂²×C₆₇	1	8	yes	complemented
240	#418	C₂²×C₆₇	1	4	yes	complemented
241	#417	C₂²×C₆₇	1	8	yes	complemented
242	#416	C₂²×C₆₇	1	8	yes	complemented
243	C₂×#102	C₂²×C₆₇	1	8	yes	complemented
244	#405	C₂²×C₆₇	1	8	yes	complemented
245	C₄×M₁₆	C₂²×C₆₇	1	4	yes	complemented
246	C₂⁵	—	2	1	no	—
247	C₂×#41	—	2	2	no	—
248	C₂×#41	—	2	8	no	—
249	C₂×#41	C₂³×C₆₇	1	4	yes	—
250	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
251	C₂²×D₄	—	2	1	no	—
252	C₂²×D₄	—	2	1	no	—
253	C₂²×D₄	—	2	2	no	—
254	C₂²×D₄	—	2	4	no	—
255	C₂²×D₄	—	2	8	no	—
256	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
257	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
258	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
259	C₂³×C₄	—	2	1	no	—
260	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
261	C₂×M₁₆	—	2	8	no	—
262	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
263	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
264	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
265	C₄⋊Q₈	—	2	8	no	—
266	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	complemented
267	#127	—	2	4	no	—
268	#127	C₂³×C₆₇	1	8	yes	—
269	#125	—	2	8	no	—
270	#125	C₂³×C₆₇	1	8	yes	complemented
271	#124	—	2	4	no	—
272	#124	C₂³×C₆₇	1	8	yes	—
273	C₂²⋊Q₈	—	2	8	no	—
274	C₂²⋊Q₈	—	2	16	no	—
275	#121	—	2	16	no	—
276	#120	—	2	16	no	—
277	C₄×Q₈	—	2	8	no	—
278	#117	—	2	2	no	—
279	#117	—	2	2	no	—
280	#117	D₄×C₆₇	1	4	yes	—
281	#117	C₂³×C₆₇	1	4	yes	—
282	#117	C₂³×C₆₇	1	4	yes	—
283	#117	C₂³×C₆₇	1	4	yes	—
284	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
285	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
286	C₂×C₂²⋊C₄	—	2	2	no	—
287	C₂×C₂²⋊C₄	—	2	4	no	—
288	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
289	C₂×C₄²	—	2	2	no	—
290	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
291	#103	C₂³×C₆₇	1	16	yes	complemented
292	#102	C₂³×C₆₇	1	32	yes	—
293	#102	C₂³×C₆₇	1	16	yes	complemented
294	#97	C₂³×C₆₇	1	8	yes	complemented
295	C₄×C₈	C₂³×C₆₇	1	8	yes	complemented
296	C₂⁴	—	2	2	no	—
297	C₂⁴	—	2	1	no	—
298	C₂⁴	—	2	4	no	—
299	C₂⁴	—	4	4	no	—
300	C₂⁴	—	8	4	no	—
301	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
302	#41	—	2	8	no	—
303	#41	—	2	8	no	—
304	#41	—	4	8	no	—
305	C₂×Q₈	—	2	1	no	—
306	C₂×Q₈	—	2	8	no	—
307	C₂×Q₈	—	2	4	no	—
308	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
309	C₂×D₄	—	2	4	no	—
310	C₂×D₄	—	2	4	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	1	no	—
313	C₂×D₄	—	2	16	no	—
314	C₂×D₄	—	2	8	no	—
315	C₂×D₄	—	2	8	no	—
316	C₂×D₄	—	2	8	no	—
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	4	4	no	—
319	C₂×D₄	—	4	4	no	—
320	C₂×D₄	—	4	8	no	—
321	C₂×D₄	—	8	8	no	—
322	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
323	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	4	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	4	4	no	—
328	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
329	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
330	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
331	C₂²×C₄	#41×C₆₇	1	8	yes	—
332	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
333	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
334	M₁₆	—	2	8	no	—
335	M₁₆	—	2	8	no	—
336	C₂×C₈	—	2	8	no	—
337	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	2	8	no	—
340	C₄⋊C₄	—	2	16	no	—
341	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
342	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
343	C₂²⋊C₄	—	2	4	no	—
344	C₂²⋊C₄	—	2	4	no	—
345	C₂²⋊C₄	—	2	16	no	—
346	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
347	C₄²	—	2	4	no	—
348	C₄²	—	2	4	no	—
349	C₄²	C₂×D₄×C₆₇	1	4	yes	—
350	C₄²	C₂⁴×C₆₇	1	4	yes	—
351	C₂³	—	2	4	no	—
352	C₂³	—	2	2	no	—
353	C₂³	—	4	2	no	—
354	C₂³	—	4	4	no	—
355	C₂³	—	4	8	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	4	4	no	—
358	C₂³	—	4	8	no	—
359	C₂³	—	8	8	no	—
360	C₂³	—	8	16	no	—
361	C₂³	#124×C₆₇	1	4	yes	—
362	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
363	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
364	Q₈	—	2	4	no	—
365	Q₈	—	4	2	no	—
366	Q₈	—	8	8	no	—
367	D₄	—	2	8	no	—
368	D₄	—	2	4	no	—
369	D₄	—	4	8	no	—
370	D₄	—	4	8	no	—
371	D₄	—	4	4	no	—
372	D₄	—	4	2	no	—
373	D₄	—	8	16	no	—
374	C₂×C₄	—	2	8	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	4	4	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	8	no	—
379	C₂×C₄	—	8	4	no	—
380	C₂×C₄	#124×C₆₇	1	4	yes	—
381	C₂×C₄	#124×C₆₇	1	8	yes	—
382	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
383	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
384	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
385	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
386	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
387	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
388	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
389	C₈	—	2	8	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	2	4	no	—
392	C₂²	—	4	4	no	—
393	C₂²	—	4	4	no	—
394	C₂²	—	4	8	no	—
395	C₂²	—	4	2	no	—
396	C₂²	—	4	4	no	—
397	C₂²	—	8	8	no	—
398	C₂²	—	8	4	no	—
399	C₂²	—	8	16	no	—
400	C₂²	C₂×#124×C₆₇	1	2	yes	—
401	C₂²	C₂×#124×C₆₇	1	4	yes	—
402	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
403	C₄	—	2	8	no	—
404	C₄	—	4	8	no	—
405	C₄	—	4	2	no	—
406	C₄	C₂×#124×C₆₇	1	4	yes	—
407	C₄	C₂×#124×C₆₇	1	2	yes	—
408	C₄	C₂²×#41×C₆₇	1	2	yes	—
409	C₂	—	2	2	no	—
410	C₂	—	2	4	no	—
411	C₂	—	8	4	no	—
412	C₂	—	8	2	no	—
413	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
414	C₂	C₆₇×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#535×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3458	C₂	semidirect
3. C₆₇×#3140	C₂	semidirect
4. C₆₇×#3138	C₂	semidirect
5. C₆₇×#3136	C₂	semidirect
6. C₆₇×#2896	C₂	semidirect
7. C₂×#418×C₆₇	C₂	semidirect
8. C₆₇×#2873	C₂	semidirect
13. #544×C₆₇	C₂²	semidirect
18. #533×C₆₇	C₂²	semidirect
25. #490×C₆₇	C₂²	semidirect
26. #489×C₆₇	C₂²	semidirect
27. #488×C₆₇	C₂²	semidirect
28. #486×C₆₇	C₂²	semidirect
29. #419×C₆₇	C₂²	semidirect
31. #418×C₆₇	C₂²	semidirect
32. #418×C₆₇	C₂²	semidirect
33. #417×C₆₇	C₂²	semidirect
34. #416×C₆₇	C₂²	semidirect
35. C₂×#102×C₆₇	C₂²	semidirect
36. #405×C₆₇	C₂²	semidirect
37. C₄×M₁₆×C₆₇	C₂²	semidirect
58. C₄⋊Q₈×C₆₇	C₂³	semidirect
62. #125×C₆₇	C₂³	semidirect
83. #103×C₆₇	C₂³	semidirect
85. #102×C₆₇	C₂³	semidirect
86. #97×C₆₇	C₂³	semidirect
87. C₄×C₈×C₆₇	C₂³	semidirect
201. #60619	C₆₇	direct (N × H)
208. #3458	C₂×C₆₇	semidirect
210. #3140	C₂×C₆₇	semidirect
211. #3138	C₂×C₆₇	semidirect
212. #3136	C₂×C₆₇	semidirect
213. #2896	C₂×C₆₇	semidirect
214. C₂×#418	C₂×C₆₇	semidirect
215. #2873	C₂×C₆₇	semidirect
216. C₆₇	#60619	direct (N × H)
221. #544	C₂²×C₆₇	semidirect
226. #533	C₂²×C₆₇	semidirect
233. #490	C₂²×C₆₇	semidirect
234. #489	C₂²×C₆₇	semidirect
235. #488	C₂²×C₆₇	semidirect
236. #486	C₂²×C₆₇	semidirect
237. #419	C₂²×C₆₇	semidirect
239. #418	C₂²×C₆₇	semidirect
240. #418	C₂²×C₆₇	semidirect
241. #417	C₂²×C₆₇	semidirect
242. #416	C₂²×C₆₇	semidirect
243. C₂×#102	C₂²×C₆₇	semidirect
244. #405	C₂²×C₆₇	semidirect
245. C₄×M₁₆	C₂²×C₆₇	semidirect
266. C₄⋊Q₈	C₂³×C₆₇	semidirect
270. #125	C₂³×C₆₇	semidirect
291. #103	C₂³×C₆₇	semidirect
293. #102	C₂³×C₆₇	semidirect
294. #97	C₂³×C₆₇	semidirect
295. C₄×C₈	C₂³×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3458	2
C₂×#535×C₆₇	2
C₆₇×#3140	2
C₆₇×#3138	2
C₆₇×#3136	2
C₆₇×#2896	2
C₂×#418×C₆₇	2
C₆₇×#2873	2
#60619	67

Subgroups — C67×#60619 #30739870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60619 #30739870