Subgroups — C₆₇×#10852 #30690103

Order 17152.

986 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#339×C₆₇	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
6	C₂×#117×C₆₇	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
8	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
9	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
10	#423×C₆₇	—	2	2	no	—
11	#423×C₆₇	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
13	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
14	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
15	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	complemented
16	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	complemented
17	C₄×C₄⋊C₄×C₆₇	C₄	1	4	yes	complemented
18	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
19	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
20	#376×C₆₇	—	2	1	no	—
21	C₂×#95×C₆₇	—	2	1	no	—
22	C₄³×C₆₇	—	2	1	no	—
23	#339×C₆₇	—	2	8	no	—
24	C₂³×C₄×C₆₇	—	2	2	no	—
25	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₆₇	—	2	2	no	—
29	C₂×M₁₆×C₆₇	—	2	4	no	—
30	C₂×M₁₆×C₆₇	—	2	4	no	—
31	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₆₇	—	2	2	no	—
33	#117×C₆₇	—	2	2	no	—
34	#117×C₆₇	—	2	2	no	—
35	#117×C₆₇	D₄	1	4	yes	—
36	#117×C₆₇	Q₈	1	4	yes	—
37	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄²×C₆₇	—	2	2	no	—
42	C₂×C₄²×C₆₇	—	2	4	no	—
43	C₂×C₄²×C₆₇	—	2	4	no	—
44	C₂×C₄²×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	—	2	2	no	—
46	C₂×C₄²×C₆₇	—	2	2	no	—
47	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	D₄	1	2	yes	—
53	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
56	C₄⋊C₈×C₆₇	—	2	4	no	—
57	C₄⋊C₈×C₆₇	—	2	4	no	—
58	#95×C₆₇	—	2	4	no	—
59	#95×C₆₇	—	2	2	no	—
60	#95×C₆₇	—	2	2	no	—
61	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	—	2	4	no	—
63	C₂²×C₄×C₆₇	—	2	8	no	—
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	4	2	no	—
67	C₂²×C₄×C₆₇	—	4	4	no	—
68	C₂²×C₄×C₆₇	—	4	4	no	—
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₆₇	#41	1	2	yes	—
86	C₂²×C₄×C₆₇	#41	1	2	yes	—
87	M₁₆×C₆₇	—	2	8	no	—
88	M₁₆×C₆₇	—	4	2	no	—
89	M₁₆×C₆₇	—	4	2	no	—
90	C₂×C₈×C₆₇	—	2	4	no	—
91	C₂×C₈×C₆₇	—	4	4	no	—
92	C₄⋊C₄×C₆₇	—	2	16	no	—
93	C₄⋊C₄×C₆₇	—	2	4	no	—
94	C₄⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	complemented
95	C₂²⋊C₄×C₆₇	—	2	4	no	—
96	C₂²⋊C₄×C₆₇	—	2	4	no	—
97	C₄²×C₆₇	—	2	4	no	—
98	C₄²×C₆₇	—	2	4	no	—
99	C₄²×C₆₇	—	2	8	no	—
100	C₄²×C₆₇	—	4	4	no	—
101	C₄²×C₆₇	—	4	4	no	—
102	C₄²×C₆₇	—	4	2	no	—
103	C₄²×C₆₇	—	4	2	no	—
104	C₄²×C₆₇	C₄²	1	2	yes	—
105	C₄²×C₆₇	C₄²	1	2	yes	—
106	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
107	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
108	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
109	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
110	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
111	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
112	C₂³×C₆₇	—	2	2	no	—
113	C₂³×C₆₇	—	2	2	no	—
114	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂³×C₆₇	#121	1	2	yes	—
118	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	4	no	—
123	C₂×C₄×C₆₇	—	2	4	no	—
124	C₂×C₄×C₆₇	—	2	8	no	—
125	C₂×C₄×C₆₇	—	2	8	no	—
126	C₂×C₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	—	4	2	no	—
128	C₂×C₄×C₆₇	—	4	16	no	—
129	C₂×C₄×C₆₇	—	4	4	no	—
130	C₂×C₄×C₆₇	—	4	4	no	—
131	C₂×C₄×C₆₇	—	4	4	no	—
132	C₂×C₄×C₆₇	—	4	4	no	—
133	C₂×C₄×C₆₇	#95	1	2	yes	—
134	C₂×C₄×C₆₇	#95	1	2	yes	—
135	C₂×C₄×C₆₇	#95	1	2	yes	—
136	C₂×C₄×C₆₇	#95	1	2	yes	—
137	C₂×C₄×C₆₇	#104	1	4	yes	—
138	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	#117	1	2	yes	—
144	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
145	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
146	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
147	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
148	C₂×C₄×C₆₇	#121	1	2	yes	—
149	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
150	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
151	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
152	C₂×C₄×C₆₇	#125	1	2	yes	—
153	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
155	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
156	C₈×C₆₇	—	4	4	no	—
157	C₂²×C₆₇	—	2	4	no	—
158	C₂²×C₆₇	—	2	4	no	—
159	C₂²×C₆₇	—	2	1	no	—
160	C₂²×C₆₇	—	2	1	no	—
161	C₂²×C₆₇	—	2	1	no	—
162	C₂²×C₆₇	—	2	1	no	—
163	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
164	C₂²×C₆₇	#380	1	2	yes	—
165	C₂²×C₆₇	#389	1	2	yes	—
166	C₂²×C₆₇	#390	1	1	yes	characteristic
167	C₂²×C₆₇	#391	1	1	yes	characteristic
168	C₂²×C₆₇	C₂×#104	1	2	yes	—
169	C₂²×C₆₇	#421	1	2	yes	—
170	C₄×C₆₇	—	2	2	no	—
171	C₄×C₆₇	—	2	2	no	—
172	C₄×C₆₇	—	2	2	no	—
173	C₄×C₆₇	—	4	8	no	—
174	C₄×C₆₇	—	4	8	no	—
175	C₄×C₆₇	#339	1	4	yes	complemented
176	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
177	C₄×C₆₇	#382	1	1	yes	characteristic
178	C₄×C₆₇	#384	1	1	yes	characteristic
179	C₄×C₆₇	#389	1	2	yes	—
180	C₄×C₆₇	#390	1	1	yes	characteristic
181	C₄×C₆₇	#391	1	1	yes	characteristic
182	#10852	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂×C₆₇	—	2	1	no	—
184	C₂×C₆₇	—	2	2	no	—
185	C₂×C₆₇	—	2	1	no	—
186	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
187	C₂×C₆₇	C₂×#339	1	1	yes	characteristic
188	C₂×C₆₇	#1733	1	1	yes	characteristic
189	C₂×C₆₇	#1806	1	4	yes	—
190	C₂×#423	C₂×C₆₇	1	1	yes	characteristic
191	C₄×#117	C₂×C₆₇	1	1	yes	characteristic
192	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
193	C₂×#339	C₂×C₆₇	1	4	yes	—
194	C₆₇	#10852	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
196	C₂×#117	C₂²×C₆₇	1	2	yes	—
197	C₂²×C₄⋊C₄	—	2	1	no	—
198	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
199	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
200	#423	—	2	2	no	—
201	#423	C₄×C₆₇	1	4	yes	complemented
202	C₂×C₄⋊C₈	—	2	1	no	—
203	C₂×C₄⋊C₈	—	2	1	no	—
204	C₄×C₄⋊C₄	—	2	2	no	—
205	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	complemented
206	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	complemented
207	C₄×C₄⋊C₄	C₄×C₆₇	1	4	yes	complemented
208	C₄×C₂²⋊C₄	—	2	1	no	—
209	C₄×C₂²⋊C₄	—	2	1	no	—
210	#376	—	2	1	no	—
211	C₂×#95	—	2	1	no	—
212	C₄³	—	2	1	no	—
213	#339	—	2	8	no	—
214	C₂³×C₄	—	2	2	no	—
215	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
216	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
217	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
218	C₂×M₁₆	—	2	2	no	—
219	C₂×M₁₆	—	2	4	no	—
220	C₂×M₁₆	—	2	4	no	—
221	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
222	C₂²×C₈	—	2	2	no	—
223	#117	—	2	2	no	—
224	#117	—	2	2	no	—
225	#117	D₄×C₆₇	1	4	yes	—
226	#117	Q₈×C₆₇	1	4	yes	—
227	C₂×C₄⋊C₄	—	2	8	no	—
228	C₂×C₄⋊C₄	—	2	4	no	—
229	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	—	2	4	no	—
233	C₂×C₄²	—	2	4	no	—
234	C₂×C₄²	—	2	2	no	—
235	C₂×C₄²	—	2	2	no	—
236	C₂×C₄²	—	2	2	no	—
237	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
238	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
239	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
240	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
241	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
242	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
243	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
244	C₂×C₄²	D₄×C₆₇	1	2	yes	—
245	C₂×C₄²	Q₈×C₆₇	1	2	yes	—
246	C₄⋊C₈	—	2	4	no	—
247	C₄⋊C₈	—	2	4	no	—
248	#95	—	2	4	no	—
249	#95	—	2	2	no	—
250	#95	—	2	2	no	—
251	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	2	8	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	—	4	4	no	—
258	C₂²×C₄	—	4	4	no	—
259	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
260	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
261	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
262	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
263	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
264	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
265	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
266	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
267	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
268	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
269	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
273	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
274	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
275	C₂²×C₄	#41×C₆₇	1	2	yes	—
276	C₂²×C₄	#41×C₆₇	1	2	yes	—
277	M₁₆	—	2	8	no	—
278	M₁₆	—	4	2	no	—
279	M₁₆	—	4	2	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	4	4	no	—
282	C₄⋊C₄	—	2	16	no	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	C₄⋊C₄×C₆₇	1	8	yes	complemented
285	C₂²⋊C₄	—	2	4	no	—
286	C₂²⋊C₄	—	2	4	no	—
287	C₄²	—	2	4	no	—
288	C₄²	—	2	4	no	—
289	C₄²	—	2	8	no	—
290	C₄²	—	4	4	no	—
291	C₄²	—	4	4	no	—
292	C₄²	—	4	2	no	—
293	C₄²	—	4	2	no	—
294	C₄²	C₄²×C₆₇	1	2	yes	—
295	C₄²	C₄²×C₆₇	1	2	yes	—
296	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
297	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
298	C₄²	C₄⋊C₄×C₆₇	1	2	yes	—
299	C₄²	C₄⋊C₄×C₆₇	1	2	yes	—
300	C₄²	C₄⋊C₄×C₆₇	1	2	yes	—
301	C₄²	C₄⋊C₄×C₆₇	1	2	yes	—
302	C₂³	—	2	2	no	—
303	C₂³	—	2	2	no	—
304	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
305	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
306	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
307	C₂³	#121×C₆₇	1	2	yes	—
308	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	8	no	—
315	C₂×C₄	—	2	8	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	16	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	#95×C₆₇	1	2	yes	—
324	C₂×C₄	#95×C₆₇	1	2	yes	—
325	C₂×C₄	#95×C₆₇	1	2	yes	—
326	C₂×C₄	#95×C₆₇	1	2	yes	—
327	C₂×C₄	#104×C₆₇	1	4	yes	—
328	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
329	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
330	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
331	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
332	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
333	C₂×C₄	#117×C₆₇	1	2	yes	—
334	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
335	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
336	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
337	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
338	C₂×C₄	#121×C₆₇	1	2	yes	—
339	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
340	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
341	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
342	C₂×C₄	#125×C₆₇	1	2	yes	—
343	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
344	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
345	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
346	C₈	—	4	4	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	4	no	—
352	C₂²	—	2	4	no	—
353	C₂²	C₂×#95×C₆₇	1	1	yes	characteristic
354	C₂²	#380×C₆₇	1	2	yes	—
355	C₂²	#389×C₆₇	1	2	yes	—
356	C₂²	#390×C₆₇	1	1	yes	characteristic
357	C₂²	#391×C₆₇	1	1	yes	characteristic
358	C₂²	C₂×#104×C₆₇	1	2	yes	—
359	C₂²	#421×C₆₇	1	2	yes	—
360	C₄	—	2	2	no	—
361	C₄	—	2	2	no	—
362	C₄	—	2	2	no	—
363	C₄	—	4	8	no	—
364	C₄	—	4	8	no	—
365	C₄	#339×C₆₇	1	4	yes	complemented
366	C₄	C₄×C₄⋊C₄×C₆₇	1	2	yes	—
367	C₄	#382×C₆₇	1	1	yes	characteristic
368	C₄	#384×C₆₇	1	1	yes	characteristic
369	C₄	#389×C₆₇	1	2	yes	—
370	C₄	#390×C₆₇	1	1	yes	characteristic
371	C₄	#391×C₆₇	1	1	yes	characteristic
372	C₂	—	2	1	no	—
373	C₂	—	2	2	no	—
374	C₂	—	2	1	no	—
375	C₂	C₆₇×C₄⋊#95	1	1	yes	characteristic
376	C₂	C₂×#339×C₆₇	1	1	yes	characteristic
377	C₂	C₆₇×#1733	1	1	yes	characteristic
378	C₂	C₆₇×#1806	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
11. #423×C₆₇	C₄	semidirect
15. C₄×C₄⋊C₄×C₆₇	C₄	semidirect
16. C₄×C₄⋊C₄×C₆₇	C₄	semidirect
17. C₄×C₄⋊C₄×C₆₇	C₄	semidirect
94. C₄⋊C₄×C₆₇	C₄⋊C₄	semidirect
175. C₄×C₆₇	#339	semidirect
182. #10852	C₆₇	direct (N × H)
194. C₆₇	#10852	direct (N × H)
201. #423	C₄×C₆₇	semidirect
205. C₄×C₄⋊C₄	C₄×C₆₇	semidirect
206. C₄×C₄⋊C₄	C₄×C₆₇	semidirect
207. C₄×C₄⋊C₄	C₄×C₆₇	semidirect
284. C₄⋊C₄	C₄⋊C₄×C₆₇	semidirect
365. C₄	#339×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#423×C₆₇	2
C₄×#117×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#339×C₆₇	2
#10852	67

Subgroups — C67×#10852 #30690103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10852 #30690103