Subgroups — C₆₇×#11100 #30690351

Order 17152.

1054 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2911	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#407×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2280	C₂	1	2	yes	maximal
4	C₆₇×#1733	C₂	1	2	yes	maximal, complemented
5	C₂×H(3,ℤ/4)×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#117×C₆₇	C₂²	1	2	yes	—
10	C₂×#117×C₆₇	C₂²	1	2	yes	—
11	#428×C₆₇	—	2	4	no	—
12	#427×C₆₇	—	2	2	no	—
13	#407×C₆₇	—	2	2	no	—
14	#407×C₆₇	C₄	1	4	yes	complemented
15	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
16	#381×C₆₇	—	2	2	no	—
17	#380×C₆₇	—	2	4	no	—
18	#379×C₆₇	—	2	2	no	—
19	C₄×C₂²⋊C₄×C₆₇	—	2	4	no	—
20	#339×C₆₇	—	2	4	no	—
21	H(3,ℤ/4)×C₆₇	—	2	2	no	—
22	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₆₇	—	2	2	no	—
24	C₂³×C₄×C₆₇	—	2	2	no	—
25	C₂³×C₄×C₆₇	—	2	1	no	—
26	C₂³×C₄×C₆₇	—	2	1	no	—
27	C₂³×C₄×C₆₇	—	2	1	no	—
28	C₂³×C₄×C₆₇	—	2	1	no	—
29	C₂³×C₄×C₆₇	—	2	2	no	—
30	C₂³×C₄×C₆₇	—	2	2	no	—
31	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
34	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	—	2	1	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
43	C₂²×C₈×C₆₇	—	2	1	no	—
44	C₂²×C₈×C₆₇	D₄	1	2	yes	—
45	#117×C₆₇	—	2	2	no	—
46	#117×C₆₇	—	2	2	no	—
47	#117×C₆₇	—	2	2	no	—
48	#117×C₆₇	—	2	2	no	—
49	#117×C₆₇	—	2	2	no	—
50	#117×C₆₇	—	2	2	no	—
51	#117×C₆₇	—	2	2	no	—
52	#117×C₆₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
55	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
56	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄²×C₆₇	—	2	2	no	—
59	C₂×C₄²×C₆₇	D₄	1	4	yes	—
60	#107×C₆₇	—	2	4	no	—
61	#106×C₆₇	—	2	4	no	—
62	C₂²⋊C₈×C₆₇	—	2	8	no	—
63	#95×C₆₇	—	2	8	no	—
64	C₂⁴×C₆₇	—	2	2	no	—
65	C₂⁴×C₆₇	—	2	1	no	—
66	C₂⁴×C₆₇	—	2	1	no	—
67	C₂⁴×C₆₇	—	2	2	no	—
68	C₂⁴×C₆₇	—	4	4	no	—
69	C₂⁴×C₆₇	C₄⋊C₄	1	2	yes	—
70	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	—	2	1	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	—	2	2	no	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	—	2	4	no	—
79	C₂²×C₄×C₆₇	—	2	2	no	—
80	C₂²×C₄×C₆₇	—	2	2	no	—
81	C₂²×C₄×C₆₇	—	2	4	no	—
82	C₂²×C₄×C₆₇	—	2	4	no	—
83	C₂²×C₄×C₆₇	—	2	4	no	—
84	C₂²×C₄×C₆₇	—	2	4	no	—
85	C₂²×C₄×C₆₇	—	4	1	no	—
86	C₂²×C₄×C₆₇	—	4	2	no	—
87	C₂²×C₄×C₆₇	—	4	2	no	—
88	C₂²×C₄×C₆₇	—	4	2	no	—
89	C₂²×C₄×C₆₇	—	4	2	no	—
90	C₂²×C₄×C₆₇	—	4	1	no	—
91	C₂²×C₄×C₆₇	—	4	2	no	—
92	C₂²×C₄×C₆₇	—	4	1	no	—
93	C₂²×C₄×C₆₇	—	4	2	no	—
94	C₂²×C₄×C₆₇	—	4	1	no	—
95	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	#41	1	2	yes	—
105	C₂²×C₄×C₆₇	#41	1	2	yes	—
106	M₁₆×C₆₇	—	2	4	no	—
107	M₁₆×C₆₇	—	2	4	no	—
108	M₁₆×C₆₇	—	2	4	no	—
109	M₁₆×C₆₇	—	4	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	4	1	no	—
114	C₂×C₈×C₆₇	—	4	1	no	—
115	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	complemented
116	C₄⋊C₄×C₆₇	—	2	4	no	—
117	C₄⋊C₄×C₆₇	—	2	4	no	—
118	C₄⋊C₄×C₆₇	—	4	2	no	—
119	C₄⋊C₄×C₆₇	—	4	2	no	—
120	C₂²⋊C₄×C₆₇	—	4	16	no	—
121	C₂²⋊C₄×C₆₇	—	4	2	no	—
122	C₂²⋊C₄×C₆₇	—	4	2	no	—
123	C₂²⋊C₄×C₆₇	—	4	2	no	—
124	C₂²⋊C₄×C₆₇	—	4	2	no	—
125	C₄²×C₆₇	—	2	4	no	—
126	C₄²×C₆₇	—	2	4	no	—
127	C₄²×C₆₇	—	4	2	no	—
128	C₄²×C₆₇	—	4	2	no	—
129	C₂³×C₆₇	—	2	8	no	—
130	C₂³×C₆₇	—	2	4	no	—
131	C₂³×C₆₇	—	2	2	no	—
132	C₂³×C₆₇	—	2	4	no	—
133	C₂³×C₆₇	—	4	4	no	—
134	C₂³×C₆₇	—	4	2	no	—
135	C₂³×C₆₇	—	4	4	no	—
136	C₂³×C₆₇	—	4	2	no	—
137	C₂³×C₆₇	—	4	4	no	—
138	C₂³×C₆₇	—	4	8	no	—
139	C₂³×C₆₇	—	4	1	no	—
140	C₂³×C₆₇	—	4	2	no	—
141	C₂³×C₆₇	—	4	1	no	—
142	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂³×C₆₇	#120	1	2	yes	—
144	C₂³×C₆₇	#121	1	1	yes	characteristic
145	C₂³×C₆₇	#121	1	1	yes	characteristic
146	C₂³×C₆₇	#123	1	2	yes	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	4	no	—
149	C₂×C₄×C₆₇	—	4	1	no	—
150	C₂×C₄×C₆₇	—	4	2	no	—
151	C₂×C₄×C₆₇	—	4	1	no	—
152	C₂×C₄×C₆₇	—	4	2	no	—
153	C₂×C₄×C₆₇	—	4	4	no	—
154	C₂×C₄×C₆₇	—	4	2	no	—
155	C₂×C₄×C₆₇	—	4	2	no	—
156	C₂×C₄×C₆₇	—	4	4	no	—
157	C₂×C₄×C₆₇	—	4	2	no	—
158	C₂×C₄×C₆₇	—	4	8	no	—
159	C₂×C₄×C₆₇	—	4	8	no	—
160	C₂×C₄×C₆₇	—	4	4	no	—
161	C₂×C₄×C₆₇	—	4	4	no	—
162	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
165	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
166	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
167	C₂×C₄×C₆₇	#121	1	2	yes	—
168	C₂×C₄×C₆₇	#121	1	2	yes	—
169	C₂×C₄×C₆₇	#121	1	2	yes	—
170	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
171	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
174	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
175	C₂×C₄×C₆₇	#123	1	2	yes	—
176	C₂×C₄×C₆₇	#124	1	2	yes	—
177	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
178	C₂×C₄×C₆₇	#126	1	2	yes	—
179	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
180	C₈×C₆₇	—	2	4	no	—
181	C₈×C₆₇	—	4	2	no	—
182	C₂²×C₆₇	—	2	4	no	—
183	C₂²×C₆₇	—	2	2	no	—
184	C₂²×C₆₇	—	2	4	no	—
185	C₂²×C₆₇	—	2	8	no	—
186	C₂²×C₆₇	—	4	1	no	—
187	C₂²×C₆₇	—	4	1	no	—
188	C₂²×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	—	4	8	no	—
190	C₂²×C₆₇	—	4	2	no	—
191	C₂²×C₆₇	—	4	4	no	—
192	C₂²×C₆₇	—	4	4	no	—
193	C₂²×C₆₇	—	4	2	no	—
194	C₂²×C₆₇	—	4	4	no	—
195	C₂²×C₆₇	#380	1	1	yes	characteristic
196	C₂²×C₆₇	#385	1	2	yes	—
197	C₂²×C₆₇	#392	1	1	yes	characteristic
198	C₂²×C₆₇	#394	1	2	yes	—
199	C₂²×C₆₇	#397	1	1	yes	characteristic
200	C₄×C₆₇	—	4	2	no	—
201	C₄×C₆₇	—	8	4	no	—
202	C₄×C₆₇	—	8	4	no	—
203	C₄×C₆₇	#386	1	1	yes	characteristic
204	C₄×C₆₇	#388	1	2	yes	—
205	C₄×C₆₇	#390	1	1	yes	characteristic
206	C₄×C₆₇	#393	1	1	yes	characteristic
207	C₄×C₆₇	#396	1	2	yes	—
208	C₄×C₆₇	#399	1	1	yes	characteristic
209	#11100	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂×C₆₇	—	2	2	no	—
211	C₂×C₆₇	—	2	1	no	—
212	C₂×C₆₇	—	2	2	no	—
213	C₂×C₆₇	—	2	1	no	—
214	C₂×C₆₇	—	4	4	no	—
215	C₂×C₆₇	#1439	1	1	yes	characteristic
216	C₂×C₆₇	#2010	1	2	yes	—
217	#2911	C₂×C₆₇	1	1	yes	characteristic, complemented
218	C₂×#407	C₂×C₆₇	1	1	yes	characteristic
219	#2280	C₂×C₆₇	1	2	yes	—
220	#1733	C₂×C₆₇	1	2	yes	complemented
221	C₂×H(3,ℤ/4)	C₂×C₆₇	1	1	yes	characteristic, complemented
222	C₆₇	#11100	1	1	yes	characteristic, direct product factor, Sylow
223	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
224	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
225	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
226	C₂×#117	C₂²×C₆₇	1	2	yes	—
227	C₂×#117	C₂²×C₆₇	1	2	yes	—
228	#428	—	2	4	no	—
229	#427	—	2	2	no	—
230	#407	—	2	2	no	—
231	#407	C₄×C₆₇	1	4	yes	complemented
232	C₂×C₂²⋊C₈	—	2	2	no	—
233	#381	—	2	2	no	—
234	#380	—	2	4	no	—
235	#379	—	2	2	no	—
236	C₄×C₂²⋊C₄	—	2	4	no	—
237	#339	—	2	4	no	—
238	H(3,ℤ/4)	—	2	2	no	—
239	C₂⁵	D₄×C₆₇	1	1	yes	characteristic
240	C₂³×C₄	—	2	2	no	—
241	C₂³×C₄	—	2	2	no	—
242	C₂³×C₄	—	2	1	no	—
243	C₂³×C₄	—	2	1	no	—
244	C₂³×C₄	—	2	1	no	—
245	C₂³×C₄	—	2	1	no	—
246	C₂³×C₄	—	2	2	no	—
247	C₂³×C₄	—	2	2	no	—
248	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
249	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
250	C₂³×C₄	Q₈×C₆₇	1	2	yes	—
251	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
252	C₂×M₁₆	—	2	2	no	—
253	C₂×M₁₆	—	2	2	no	—
254	C₂×M₁₆	—	2	1	no	—
255	C₂×M₁₆	—	2	2	no	—
256	C₂×M₁₆	—	2	2	no	—
257	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
258	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
259	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
260	C₂²×C₈	—	2	1	no	—
261	C₂²×C₈	D₄×C₆₇	1	2	yes	—
262	#117	—	2	2	no	—
263	#117	—	2	2	no	—
264	#117	—	2	2	no	—
265	#117	—	2	2	no	—
266	#117	—	2	2	no	—
267	#117	—	2	2	no	—
268	#117	—	2	2	no	—
269	#117	—	2	2	no	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
272	C₂×C₂²⋊C₄	—	2	8	no	—
273	C₂×C₂²⋊C₄	—	2	2	no	—
274	C₂×C₂²⋊C₄	—	2	2	no	—
275	C₂×C₄²	—	2	2	no	—
276	C₂×C₄²	D₄×C₆₇	1	4	yes	—
277	#107	—	2	4	no	—
278	#106	—	2	4	no	—
279	C₂²⋊C₈	—	2	8	no	—
280	#95	—	2	8	no	—
281	C₂⁴	—	2	2	no	—
282	C₂⁴	—	2	1	no	—
283	C₂⁴	—	2	1	no	—
284	C₂⁴	—	2	2	no	—
285	C₂⁴	—	4	4	no	—
286	C₂⁴	C₄⋊C₄×C₆₇	1	2	yes	—
287	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	2	1	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	4	no	—
299	C₂²×C₄	—	2	4	no	—
300	C₂²×C₄	—	2	4	no	—
301	C₂²×C₄	—	2	4	no	—
302	C₂²×C₄	—	4	1	no	—
303	C₂²×C₄	—	4	2	no	—
304	C₂²×C₄	—	4	2	no	—
305	C₂²×C₄	—	4	2	no	—
306	C₂²×C₄	—	4	2	no	—
307	C₂²×C₄	—	4	1	no	—
308	C₂²×C₄	—	4	2	no	—
309	C₂²×C₄	—	4	1	no	—
310	C₂²×C₄	—	4	2	no	—
311	C₂²×C₄	—	4	1	no	—
312	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
313	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
314	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
315	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
316	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
317	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
318	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
319	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
320	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
321	C₂²×C₄	#41×C₆₇	1	2	yes	—
322	C₂²×C₄	#41×C₆₇	1	2	yes	—
323	M₁₆	—	2	4	no	—
324	M₁₆	—	2	4	no	—
325	M₁₆	—	2	4	no	—
326	M₁₆	—	4	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	4	1	no	—
331	C₂×C₈	—	4	1	no	—
332	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	complemented
333	C₄⋊C₄	—	2	4	no	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	4	2	no	—
336	C₄⋊C₄	—	4	2	no	—
337	C₂²⋊C₄	—	4	16	no	—
338	C₂²⋊C₄	—	4	2	no	—
339	C₂²⋊C₄	—	4	2	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₄²	—	2	4	no	—
343	C₄²	—	2	4	no	—
344	C₄²	—	4	2	no	—
345	C₄²	—	4	2	no	—
346	C₂³	—	2	8	no	—
347	C₂³	—	2	4	no	—
348	C₂³	—	2	2	no	—
349	C₂³	—	2	4	no	—
350	C₂³	—	4	4	no	—
351	C₂³	—	4	2	no	—
352	C₂³	—	4	4	no	—
353	C₂³	—	4	2	no	—
354	C₂³	—	4	4	no	—
355	C₂³	—	4	8	no	—
356	C₂³	—	4	1	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	4	1	no	—
359	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
360	C₂³	#120×C₆₇	1	2	yes	—
361	C₂³	#121×C₆₇	1	1	yes	characteristic
362	C₂³	#121×C₆₇	1	1	yes	characteristic
363	C₂³	#123×C₆₇	1	2	yes	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	4	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	4	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	8	no	—
376	C₂×C₄	—	4	8	no	—
377	C₂×C₄	—	4	4	no	—
378	C₂×C₄	—	4	4	no	—
379	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
380	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
381	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
382	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
383	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
384	C₂×C₄	#121×C₆₇	1	2	yes	—
385	C₂×C₄	#121×C₆₇	1	2	yes	—
386	C₂×C₄	#121×C₆₇	1	2	yes	—
387	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
388	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
389	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
390	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
391	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
392	C₂×C₄	#123×C₆₇	1	2	yes	—
393	C₂×C₄	#124×C₆₇	1	2	yes	—
394	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
395	C₂×C₄	#126×C₆₇	1	2	yes	—
396	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
397	C₈	—	2	4	no	—
398	C₈	—	4	2	no	—
399	C₂²	—	2	4	no	—
400	C₂²	—	2	2	no	—
401	C₂²	—	2	4	no	—
402	C₂²	—	2	8	no	—
403	C₂²	—	4	1	no	—
404	C₂²	—	4	1	no	—
405	C₂²	—	4	2	no	—
406	C₂²	—	4	8	no	—
407	C₂²	—	4	2	no	—
408	C₂²	—	4	4	no	—
409	C₂²	—	4	4	no	—
410	C₂²	—	4	4	no	—
411	C₂²	—	4	2	no	—
412	C₂²	#380×C₆₇	1	1	yes	characteristic
413	C₂²	#385×C₆₇	1	2	yes	—
414	C₂²	#392×C₆₇	1	1	yes	characteristic
415	C₂²	#394×C₆₇	1	2	yes	—
416	C₂²	#397×C₆₇	1	1	yes	characteristic
417	C₄	—	4	2	no	—
418	C₄	—	8	4	no	—
419	C₄	—	8	4	no	—
420	C₄	#386×C₆₇	1	1	yes	characteristic
421	C₄	#388×C₆₇	1	2	yes	—
422	C₄	#390×C₆₇	1	1	yes	characteristic
423	C₄	#393×C₆₇	1	1	yes	characteristic
424	C₄	#396×C₆₇	1	2	yes	—
425	C₄	#399×C₆₇	1	1	yes	characteristic
426	C₂	—	2	2	no	—
427	C₂	—	2	1	no	—
428	C₂	—	2	2	no	—
429	C₂	—	2	1	no	—
430	C₂	—	4	4	no	—
431	C₂	C₆₇×#1439	1	1	yes	characteristic
432	C₂	C₆₇×#2010	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	#127
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
socle	C₂²×C₆₇	#392
prime radical	C₂⁵×C₆₇	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2911	C₂	semidirect
4. C₆₇×#1733	C₂	semidirect
5. C₂×H(3,ℤ/4)×C₆₇	C₂	semidirect
14. #407×C₆₇	C₄	semidirect
115. C₂×C₈×C₆₇	C₂²⋊C₄	semidirect
209. #11100	C₆₇	direct (N × H)
217. #2911	C₂×C₆₇	semidirect
220. #1733	C₂×C₆₇	semidirect
221. C₂×H(3,ℤ/4)	C₂×C₆₇	semidirect
222. C₆₇	#11100	direct (N × H)
231. #407	C₄×C₆₇	semidirect
332. C₂×C₈	C₂²⋊C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2911	2
C₂×#407×C₆₇	2
C₆₇×#2280	2
C₆₇×#1733	2
C₂×H(3,ℤ/4)×C₆₇	2
#11100	67

Subgroups — C67×#11100 #30690351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#11100 #30690351