Subgroups — C₆₇×#11097 #30690348

Order 17152.

1006 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	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#407×C₆₇	C₂	1	2	yes	maximal
3	C₂²×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#341×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#341×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#108×C₆₇	—	2	4	no	—
12	C₂×#108×C₆₇	—	2	2	no	—
13	#407×C₆₇	—	2	4	no	—
14	#407×C₆₇	—	2	4	no	—
15	C₂×C₂²⋊C₈×C₆₇	—	2	4	no	—
16	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
17	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
18	C₂×C₂²⋊C₈×C₆₇	C₄	1	4	yes	—
19	#341×C₆₇	—	2	4	no	—
20	#341×C₆₇	—	2	2	no	—
21	C₂⁵×C₆₇	Q₈	1	1	yes	characteristic
22	C₂³×C₄×C₆₇	—	2	1	no	—
23	C₂³×C₄×C₆₇	—	2	1	no	—
24	C₂³×C₄×C₆₇	—	2	2	no	—
25	C₂³×C₄×C₆₇	—	2	2	no	—
26	C₂³×C₄×C₆₇	—	2	1	no	—
27	C₂³×C₄×C₆₇	—	2	1	no	—
28	C₂³×C₄×C₆₇	—	2	2	no	—
29	C₂³×C₄×C₆₇	—	2	2	no	—
30	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₆₇	D₄	1	2	yes	—
32	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	—	2	2	no	—
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
45	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
46	C₂²×C₈×C₆₇	—	2	2	no	—
47	C₂²×C₈×C₆₇	—	2	2	no	—
48	C₂²×C₈×C₆₇	—	2	2	no	—
49	C₂²×C₈×C₆₇	—	2	1	no	—
50	C₂²×C₈×C₆₇	—	2	1	no	—
51	C₂²×C₈×C₆₇	—	2	2	no	—
52	C₂²×C₈×C₆₇	—	2	2	no	—
53	C₂²×C₈×C₆₇	—	2	2	no	—
54	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
56	C₂²×C₈×C₆₇	D₄	1	2	yes	—
57	C₂²×C₈×C₆₇	D₄	1	2	yes	—
58	#108×C₆₇	—	2	8	no	—
59	C₂²⋊C₈×C₆₇	—	2	16	no	—
60	C₂²⋊C₈×C₆₇	—	2	8	no	—
61	C₂⁴×C₆₇	—	2	4	no	—
62	C₂⁴×C₆₇	—	2	1	no	—
63	C₂⁴×C₆₇	—	2	1	no	—
64	C₂⁴×C₆₇	—	2	2	no	—
65	C₂⁴×C₆₇	—	2	2	no	—
66	C₂⁴×C₆₇	—	4	2	no	—
67	C₂⁴×C₆₇	C₄⋊C₄	1	2	yes	—
68	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	—	2	1	no	—
70	C₂²×C₄×C₆₇	—	2	1	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	—	2	4	no	—
76	C₂²×C₄×C₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	—	2	2	no	—
79	C₂²×C₄×C₆₇	—	2	2	no	—
80	C₂²×C₄×C₆₇	—	2	4	no	—
81	C₂²×C₄×C₆₇	—	4	1	no	—
82	C₂²×C₄×C₆₇	—	4	2	no	—
83	C₂²×C₄×C₆₇	—	4	2	no	—
84	C₂²×C₄×C₆₇	—	4	2	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	1	no	—
89	C₂²×C₄×C₆₇	—	4	2	no	—
90	C₂²×C₄×C₆₇	—	4	1	no	—
91	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	#41	1	2	yes	—
101	C₂²×C₄×C₆₇	#41	1	2	yes	—
102	M₁₆×C₆₇	—	2	4	no	—
103	M₁₆×C₆₇	—	2	4	no	—
104	M₁₆×C₆₇	—	2	4	no	—
105	M₁₆×C₆₇	—	2	4	no	—
106	M₁₆×C₆₇	—	4	2	no	—
107	M₁₆×C₆₇	—	4	2	no	—
108	M₁₆×C₆₇	—	4	2	no	—
109	M₁₆×C₆₇	—	4	2	no	—
110	C₂×C₈×C₆₇	—	2	4	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	4	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	4	no	—
116	C₂×C₈×C₆₇	—	2	4	no	—
117	C₂×C₈×C₆₇	—	2	4	no	—
118	C₂×C₈×C₆₇	—	4	2	no	—
119	C₂×C₈×C₆₇	—	4	2	no	—
120	C₂×C₈×C₆₇	—	4	2	no	—
121	C₂×C₈×C₆₇	—	4	2	no	—
122	C₂×C₈×C₆₇	—	4	2	no	—
123	C₂×C₈×C₆₇	—	4	1	no	—
124	C₂×C₈×C₆₇	—	4	1	no	—
125	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
126	C₂³×C₆₇	—	2	2	no	—
127	C₂³×C₆₇	—	2	2	no	—
128	C₂³×C₆₇	—	2	4	no	—
129	C₂³×C₆₇	—	2	4	no	—
130	C₂³×C₆₇	—	2	2	no	—
131	C₂³×C₆₇	—	2	4	no	—
132	C₂³×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	—	2	4	no	—
134	C₂³×C₆₇	—	4	8	no	—
135	C₂³×C₆₇	—	4	8	no	—
136	C₂³×C₆₇	—	4	2	no	—
137	C₂³×C₆₇	—	4	2	no	—
138	C₂³×C₆₇	—	4	1	no	—
139	C₂³×C₆₇	—	4	2	no	—
140	C₂³×C₆₇	—	4	1	no	—
141	C₂³×C₆₇	#108	1	4	yes	complemented
142	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
144	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
145	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
146	C₂³×C₆₇	C₂²⋊Q₈	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
161	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
162	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
163	C₂×C₄×C₆₇	#120	1	2	yes	—
164	C₂×C₄×C₆₇	#121	1	2	yes	—
165	C₂×C₄×C₆₇	#121	1	2	yes	—
166	C₂×C₄×C₆₇	#121	1	2	yes	—
167	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
168	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
169	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
170	C₂×C₄×C₆₇	#123	1	2	yes	—
171	C₂×C₄×C₆₇	#123	1	2	yes	—
172	C₂×C₄×C₆₇	#124	1	2	yes	—
173	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
174	C₂×C₄×C₆₇	#126	1	2	yes	—
175	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
176	C₈×C₆₇	—	2	4	no	—
177	C₈×C₆₇	—	4	2	no	—
178	C₈×C₆₇	—	4	4	no	—
179	C₈×C₆₇	—	4	4	no	—
180	C₂²×C₆₇	—	2	1	no	—
181	C₂²×C₆₇	—	2	1	no	—
182	C₂²×C₆₇	—	2	1	no	—
183	C₂²×C₆₇	—	2	1	no	—
184	C₂²×C₆₇	—	2	4	no	—
185	C₂²×C₆₇	—	2	4	no	—
186	C₂²×C₆₇	—	2	4	no	—
187	C₂²×C₆₇	—	2	8	no	—
188	C₂²×C₆₇	—	4	4	no	—
189	C₂²×C₆₇	—	4	8	no	—
190	C₂²×C₆₇	—	4	4	no	—
191	C₂²×C₆₇	—	4	2	no	—
192	C₂²×C₆₇	—	4	4	no	—
193	C₂²×C₆₇	—	4	2	no	—
194	C₂²×C₆₇	#380	1	1	yes	characteristic
195	C₂²×C₆₇	#385	1	2	yes	—
196	C₂²×C₆₇	#393	1	1	yes	characteristic
197	C₂²×C₆₇	#396	1	2	yes	—
198	C₂²×C₆₇	#399	1	1	yes	characteristic
199	C₂²×C₆₇	C₂×#108	1	2	yes	—
200	C₂²×C₆₇	#430	1	2	yes	—
201	C₄×C₆₇	—	4	2	no	—
202	C₄×C₆₇	#386	1	1	yes	characteristic
203	C₄×C₆₇	#388	1	2	yes	—
204	C₄×C₆₇	#390	1	1	yes	characteristic
205	C₄×C₆₇	#392	1	1	yes	characteristic
206	C₄×C₆₇	#394	1	2	yes	—
207	C₄×C₆₇	#397	1	1	yes	characteristic
208	#11097	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂×C₆₇	—	2	1	no	—
210	C₂×C₆₇	—	2	2	no	—
211	C₂×C₆₇	—	2	1	no	—
212	C₂×C₆₇	—	4	4	no	—
213	C₂×C₆₇	#1439	1	1	yes	characteristic
214	C₂×C₆₇	#1810	1	1	yes	characteristic
215	C₂×C₆₇	#1811	1	1	yes	characteristic
216	C₂×C₆₇	#1855	1	4	yes	—
217	C₂²×#108	C₂×C₆₇	1	1	yes	characteristic, complemented
218	C₂×#407	C₂×C₆₇	1	2	yes	—
219	C₂²×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic
220	C₂×#341	C₂×C₆₇	1	2	yes	complemented
221	C₂×#341	C₂×C₆₇	1	1	yes	characteristic, complemented
222	C₆₇	#11097	1	1	yes	characteristic, direct product factor, Sylow
223	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
224	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
225	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
226	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
227	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
228	C₂×#108	—	2	4	no	—
229	C₂×#108	—	2	2	no	—
230	#407	—	2	4	no	—
231	#407	—	2	4	no	—
232	C₂×C₂²⋊C₈	—	2	4	no	—
233	C₂×C₂²⋊C₈	—	2	2	no	—
234	C₂×C₂²⋊C₈	—	2	2	no	—
235	C₂×C₂²⋊C₈	C₄×C₆₇	1	4	yes	—
236	#341	—	2	4	no	—
237	#341	—	2	2	no	—
238	C₂⁵	Q₈×C₆₇	1	1	yes	characteristic
239	C₂³×C₄	—	2	1	no	—
240	C₂³×C₄	—	2	1	no	—
241	C₂³×C₄	—	2	2	no	—
242	C₂³×C₄	—	2	2	no	—
243	C₂³×C₄	—	2	1	no	—
244	C₂³×C₄	—	2	1	no	—
245	C₂³×C₄	—	2	2	no	—
246	C₂³×C₄	—	2	2	no	—
247	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
248	C₂³×C₄	D₄×C₆₇	1	2	yes	—
249	C₂³×C₄	Q₈×C₆₇	1	1	yes	characteristic
250	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
251	C₂×M₁₆	—	2	2	no	—
252	C₂×M₁₆	—	2	2	no	—
253	C₂×M₁₆	—	2	2	no	—
254	C₂×M₁₆	—	2	2	no	—
255	C₂×M₁₆	—	2	2	no	—
256	C₂×M₁₆	—	2	2	no	—
257	C₂×M₁₆	—	2	2	no	—
258	C₂×M₁₆	—	2	2	no	—
259	C₂×M₁₆	—	2	2	no	—
260	C₂×M₁₆	—	2	2	no	—
261	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
262	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
263	C₂²×C₈	—	2	2	no	—
264	C₂²×C₈	—	2	2	no	—
265	C₂²×C₈	—	2	2	no	—
266	C₂²×C₈	—	2	1	no	—
267	C₂²×C₈	—	2	1	no	—
268	C₂²×C₈	—	2	2	no	—
269	C₂²×C₈	—	2	2	no	—
270	C₂²×C₈	—	2	2	no	—
271	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
272	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
273	C₂²×C₈	D₄×C₆₇	1	2	yes	—
274	C₂²×C₈	D₄×C₆₇	1	2	yes	—
275	#108	—	2	8	no	—
276	C₂²⋊C₈	—	2	16	no	—
277	C₂²⋊C₈	—	2	8	no	—
278	C₂⁴	—	2	4	no	—
279	C₂⁴	—	2	1	no	—
280	C₂⁴	—	2	1	no	—
281	C₂⁴	—	2	2	no	—
282	C₂⁴	—	2	2	no	—
283	C₂⁴	—	4	2	no	—
284	C₂⁴	C₄⋊C₄×C₆₇	1	2	yes	—
285	C₂⁴	C₂×Q₈×C₆₇	1	1	yes	characteristic
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	4	no	—
293	C₂²×C₄	—	2	4	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	4	no	—
298	C₂²×C₄	—	4	1	no	—
299	C₂²×C₄	—	4	2	no	—
300	C₂²×C₄	—	4	2	no	—
301	C₂²×C₄	—	4	2	no	—
302	C₂²×C₄	—	4	1	no	—
303	C₂²×C₄	—	4	2	no	—
304	C₂²×C₄	—	4	2	no	—
305	C₂²×C₄	—	4	1	no	—
306	C₂²×C₄	—	4	2	no	—
307	C₂²×C₄	—	4	1	no	—
308	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
309	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
310	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
311	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
312	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
313	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
314	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
315	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
316	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
317	C₂²×C₄	#41×C₆₇	1	2	yes	—
318	C₂²×C₄	#41×C₆₇	1	2	yes	—
319	M₁₆	—	2	4	no	—
320	M₁₆	—	2	4	no	—
321	M₁₆	—	2	4	no	—
322	M₁₆	—	2	4	no	—
323	M₁₆	—	4	2	no	—
324	M₁₆	—	4	2	no	—
325	M₁₆	—	4	2	no	—
326	M₁₆	—	4	2	no	—
327	C₂×C₈	—	2	4	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	4	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	4	no	—
333	C₂×C₈	—	2	4	no	—
334	C₂×C₈	—	2	4	no	—
335	C₂×C₈	—	4	2	no	—
336	C₂×C₈	—	4	2	no	—
337	C₂×C₈	—	4	2	no	—
338	C₂×C₈	—	4	2	no	—
339	C₂×C₈	—	4	2	no	—
340	C₂×C₈	—	4	1	no	—
341	C₂×C₈	—	4	1	no	—
342	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
343	C₂³	—	2	2	no	—
344	C₂³	—	2	2	no	—
345	C₂³	—	2	4	no	—
346	C₂³	—	2	4	no	—
347	C₂³	—	2	2	no	—
348	C₂³	—	2	4	no	—
349	C₂³	—	2	2	no	—
350	C₂³	—	2	4	no	—
351	C₂³	—	4	8	no	—
352	C₂³	—	4	8	no	—
353	C₂³	—	4	2	no	—
354	C₂³	—	4	2	no	—
355	C₂³	—	4	1	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	4	1	no	—
358	C₂³	#108×C₆₇	1	4	yes	complemented
359	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
360	C₂³	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
361	C₂³	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
362	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
363	C₂³	C₂²⋊Q₈×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	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
376	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
377	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
378	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
379	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
380	C₂×C₄	#120×C₆₇	1	2	yes	—
381	C₂×C₄	#121×C₆₇	1	2	yes	—
382	C₂×C₄	#121×C₆₇	1	2	yes	—
383	C₂×C₄	#121×C₆₇	1	2	yes	—
384	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
385	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
386	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
387	C₂×C₄	#123×C₆₇	1	2	yes	—
388	C₂×C₄	#123×C₆₇	1	2	yes	—
389	C₂×C₄	#124×C₆₇	1	2	yes	—
390	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
391	C₂×C₄	#126×C₆₇	1	2	yes	—
392	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
393	C₈	—	2	4	no	—
394	C₈	—	4	2	no	—
395	C₈	—	4	4	no	—
396	C₈	—	4	4	no	—
397	C₂²	—	2	1	no	—
398	C₂²	—	2	1	no	—
399	C₂²	—	2	1	no	—
400	C₂²	—	2	1	no	—
401	C₂²	—	2	4	no	—
402	C₂²	—	2	4	no	—
403	C₂²	—	2	4	no	—
404	C₂²	—	2	8	no	—
405	C₂²	—	4	4	no	—
406	C₂²	—	4	8	no	—
407	C₂²	—	4	4	no	—
408	C₂²	—	4	2	no	—
409	C₂²	—	4	4	no	—
410	C₂²	—	4	2	no	—
411	C₂²	#380×C₆₇	1	1	yes	characteristic
412	C₂²	#385×C₆₇	1	2	yes	—
413	C₂²	#393×C₆₇	1	1	yes	characteristic
414	C₂²	#396×C₆₇	1	2	yes	—
415	C₂²	#399×C₆₇	1	1	yes	characteristic
416	C₂²	C₂×#108×C₆₇	1	2	yes	—
417	C₂²	#430×C₆₇	1	2	yes	—
418	C₄	—	4	2	no	—
419	C₄	#386×C₆₇	1	1	yes	characteristic
420	C₄	#388×C₆₇	1	2	yes	—
421	C₄	#390×C₆₇	1	1	yes	characteristic
422	C₄	#392×C₆₇	1	1	yes	characteristic
423	C₄	#394×C₆₇	1	2	yes	—
424	C₄	#397×C₆₇	1	1	yes	characteristic
425	C₂	—	2	1	no	—
426	C₂	—	2	2	no	—
427	C₂	—	2	1	no	—
428	C₂	—	4	4	no	—
429	C₂	C₆₇×#1439	1	1	yes	characteristic
430	C₂	C₆₇×#1810	1	1	yes	characteristic
431	C₂	C₆₇×#1811	1	1	yes	characteristic
432	C₂	C₆₇×#1855	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁵×C₆₇	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#108×C₆₇	C₂	semidirect
4. C₂×#341×C₆₇	C₂	semidirect
5. C₂×#341×C₆₇	C₂	semidirect
141. C₂³×C₆₇	#108	semidirect
208. #11097	C₆₇	direct (N × H)
217. C₂²×#108	C₂×C₆₇	semidirect
220. C₂×#341	C₂×C₆₇	semidirect
221. C₂×#341	C₂×C₆₇	semidirect
222. C₆₇	#11097	direct (N × H)
358. C₂³	#108×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#108×C₆₇	2
C₂×#407×C₆₇	2
C₂²×C₂²⋊C₈×C₆₇	2
C₂×#341×C₆₇	2
#11097	67

Subgroups — C67×#11097 #30690348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#11097 #30690348