Subgroups — C₆₇×#10851 #30690102

Order 17152.

978 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2285	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	C₄×M₁₆×C₆₇	—	2	2	no	—
11	C₄×M₁₆×C₆₇	C₄	1	4	yes	complemented
12	C₂×#97×C₆₇	—	2	1	no	—
13	C₂×C₄×C₈×C₆₇	—	2	1	no	—
14	#384×C₆₇	C₄	1	2	yes	complemented
15	#382×C₆₇	C₄	1	2	yes	complemented
16	#380×C₆₇	—	2	1	no	—
17	#380×C₆₇	—	2	1	no	—
18	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
19	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
20	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
21	C₄×C₄⋊C₄×C₆₇	C₄	1	4	yes	complemented
22	C₂×#95×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	2	no	—
30	C₂×M₁₆×C₆₇	—	2	2	no	—
31	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
33	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
34	C₂²×C₈×C₆₇	—	2	2	no	—
35	#117×C₆₇	—	2	4	no	—
36	#117×C₆₇	—	2	4	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄²×C₆₇	—	2	4	no	—
44	C₂×C₄²×C₆₇	—	2	4	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	2	yes	—
50	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
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	#97×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	2	no	—
69	C₂²×C₄×C₆₇	—	4	2	no	—
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₆₇	#41	1	2	yes	—
87	C₂²×C₄×C₆₇	#41	1	2	yes	—
88	M₁₆×C₆₇	—	2	4	no	—
89	M₁₆×C₆₇	—	2	4	no	—
90	M₁₆×C₆₇	—	2	4	no	—
91	M₁₆×C₆₇	C₄⋊C₄	1	8	yes	complemented
92	C₂×C₈×C₆₇	—	2	4	no	—
93	C₂×C₈×C₆₇	—	2	4	no	—
94	C₂×C₈×C₆₇	—	2	4	no	—
95	C₄⋊C₄×C₆₇	—	2	16	no	—
96	C₄⋊C₄×C₆₇	—	2	8	no	—
97	C₄⋊C₄×C₆₇	—	4	4	no	—
98	C₄⋊C₄×C₆₇	—	4	4	no	—
99	C₂²⋊C₄×C₆₇	—	4	2	no	—
100	C₂²⋊C₄×C₆₇	—	4	2	no	—
101	C₄²×C₆₇	—	2	8	no	—
102	C₄²×C₆₇	—	4	4	no	—
103	C₄²×C₆₇	—	4	4	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	#124	1	1	yes	characteristic
118	C₂³×C₆₇	#124	1	1	yes	characteristic
119	C₂³×C₆₇	#127	1	1	yes	characteristic
120	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	4	no	—
125	C₂×C₄×C₆₇	—	2	4	no	—
126	C₂×C₄×C₆₇	—	2	8	no	—
127	C₂×C₄×C₆₇	—	4	2	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	—	4	8	no	—
130	C₂×C₄×C₆₇	—	4	8	no	—
131	C₂×C₄×C₆₇	—	4	4	no	—
132	C₂×C₄×C₆₇	—	4	4	no	—
133	C₂×C₄×C₆₇	—	4	4	no	—
134	C₂×C₄×C₆₇	—	4	4	no	—
135	C₂×C₄×C₆₇	—	4	4	no	—
136	C₂×C₄×C₆₇	#95	1	2	yes	—
137	C₂×C₄×C₆₇	#95	1	2	yes	—
138	C₂×C₄×C₆₇	#95	1	2	yes	—
139	C₂×C₄×C₆₇	#95	1	2	yes	—
140	C₂×C₄×C₆₇	#104	1	4	yes	—
141	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	#117	1	2	yes	—
147	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
148	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
149	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
150	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
151	C₂×C₄×C₆₇	#121	1	2	yes	—
152	C₂×C₄×C₆₇	#121	1	2	yes	—
153	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
154	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
155	C₂×C₄×C₆₇	#125	1	2	yes	—
156	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
157	C₈×C₆₇	—	2	8	no	—
158	C₂²×C₆₇	—	2	2	no	—
159	C₂²×C₆₇	—	2	2	no	—
160	C₂²×C₆₇	—	2	2	no	—
161	C₂²×C₆₇	—	2	2	no	—
162	C₂²×C₆₇	—	2	1	no	—
163	C₂²×C₆₇	—	2	1	no	—
164	C₂²×C₆₇	—	2	1	no	—
165	C₂²×C₆₇	—	2	1	no	—
166	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
167	C₂²×C₆₇	#382	1	1	yes	characteristic
168	C₂²×C₆₇	#384	1	1	yes	characteristic
169	C₂²×C₆₇	#390	1	1	yes	characteristic
170	C₂²×C₆₇	#390	1	1	yes	characteristic
171	C₂²×C₆₇	#391	1	1	yes	characteristic
172	C₂²×C₆₇	#391	1	1	yes	characteristic
173	C₂²×C₆₇	C₂×#104	1	2	yes	—
174	C₂²×C₆₇	#421	1	2	yes	—
175	C₄×C₆₇	—	2	2	no	—
176	C₄×C₆₇	—	2	2	no	—
177	C₄×C₆₇	—	2	2	no	—
178	C₄×C₆₇	—	4	8	no	—
179	C₄×C₆₇	—	8	4	no	—
180	C₄×C₆₇	#339	1	4	yes	complemented
181	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
182	C₄×C₆₇	#380	1	2	yes	—
183	C₄×C₆₇	#389	1	2	yes	—
184	C₄×C₆₇	#389	1	2	yes	—
185	#10851	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂×C₆₇	—	2	1	no	—
187	C₂×C₆₇	—	2	1	no	—
188	C₂×C₆₇	—	2	1	no	—
189	C₂×C₆₇	—	2	1	no	—
190	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
191	C₂×C₆₇	C₂×#339	1	1	yes	characteristic
192	C₂×C₆₇	#1733	1	1	yes	characteristic
193	C₂×C₆₇	#1980	1	2	yes	—
194	C₂×C₆₇	#1986	1	2	yes	—
195	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic
196	#2285	C₂×C₆₇	1	1	yes	characteristic
197	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
198	C₂×#339	C₂×C₆₇	1	4	yes	—
199	C₆₇	#10851	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
201	C₂×#117	C₂²×C₆₇	1	2	yes	—
202	C₂²×C₄⋊C₄	—	2	1	no	—
203	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
204	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
205	C₄×M₁₆	—	2	2	no	—
206	C₄×M₁₆	C₄×C₆₇	1	4	yes	complemented
207	C₂×#97	—	2	1	no	—
208	C₂×C₄×C₈	—	2	1	no	—
209	#384	C₄×C₆₇	1	2	yes	complemented
210	#382	C₄×C₆₇	1	2	yes	complemented
211	#380	—	2	1	no	—
212	#380	—	2	1	no	—
213	C₄×C₄⋊C₄	—	2	2	no	—
214	C₄×C₄⋊C₄	—	2	1	no	—
215	C₄×C₄⋊C₄	—	2	1	no	—
216	C₄×C₄⋊C₄	C₄×C₆₇	1	4	yes	complemented
217	C₂×#95	—	2	1	no	—
218	#339	—	2	8	no	—
219	C₂³×C₄	—	2	2	no	—
220	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
221	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
222	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
223	C₂×M₁₆	—	2	2	no	—
224	C₂×M₁₆	—	2	2	no	—
225	C₂×M₁₆	—	2	2	no	—
226	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
227	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
228	C₂×M₁₆	Q₈×C₆₇	1	4	yes	—
229	C₂²×C₈	—	2	2	no	—
230	#117	—	2	4	no	—
231	#117	—	2	4	no	—
232	C₂×C₄⋊C₄	—	2	8	no	—
233	C₂×C₄⋊C₄	—	2	4	no	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	C₂×C₄⋊C₄	—	2	2	no	—
236	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
237	C₂×C₂²⋊C₄	—	2	2	no	—
238	C₂×C₄²	—	2	4	no	—
239	C₂×C₄²	—	2	4	no	—
240	C₂×C₄²	—	2	2	no	—
241	C₂×C₄²	—	2	2	no	—
242	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
243	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
244	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
245	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
246	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
247	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
248	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
249	C₂×C₄²	D₄×C₆₇	1	2	yes	—
250	C₂×C₄²	Q₈×C₆₇	1	2	yes	—
251	#97	—	2	4	no	—
252	C₄×C₈	—	2	4	no	—
253	#95	—	2	4	no	—
254	#95	—	2	2	no	—
255	#95	—	2	2	no	—
256	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	2	8	no	—
259	C₂²×C₄	—	2	4	no	—
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	4	4	no	—
263	C₂²×C₄	—	4	2	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
266	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
267	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
268	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
269	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
270	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
271	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
272	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
273	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
274	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
275	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
277	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
279	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
280	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
281	C₂²×C₄	#41×C₆₇	1	2	yes	—
282	C₂²×C₄	#41×C₆₇	1	2	yes	—
283	M₁₆	—	2	4	no	—
284	M₁₆	—	2	4	no	—
285	M₁₆	—	2	4	no	—
286	M₁₆	C₄⋊C₄×C₆₇	1	8	yes	complemented
287	C₂×C₈	—	2	4	no	—
288	C₂×C₈	—	2	4	no	—
289	C₂×C₈	—	2	4	no	—
290	C₄⋊C₄	—	2	16	no	—
291	C₄⋊C₄	—	2	8	no	—
292	C₄⋊C₄	—	4	4	no	—
293	C₄⋊C₄	—	4	4	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₄²	—	2	8	no	—
297	C₄²	—	4	4	no	—
298	C₄²	—	4	4	no	—
299	C₄²	C₄²×C₆₇	1	2	yes	—
300	C₄²	C₄²×C₆₇	1	2	yes	—
301	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
302	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
303	C₄²	C₄⋊C₄×C₆₇	1	2	yes	—
304	C₄²	C₄⋊C₄×C₆₇	1	2	yes	—
305	C₄²	C₄⋊C₄×C₆₇	1	2	yes	—
306	C₄²	C₄⋊C₄×C₆₇	1	2	yes	—
307	C₂³	—	2	2	no	—
308	C₂³	—	2	2	no	—
309	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
310	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
311	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
312	C₂³	#124×C₆₇	1	1	yes	characteristic
313	C₂³	#124×C₆₇	1	1	yes	characteristic
314	C₂³	#127×C₆₇	1	1	yes	characteristic
315	C₂³	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	8	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	8	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	4	8	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	—	4	4	no	—
330	C₂×C₄	—	4	4	no	—
331	C₂×C₄	#95×C₆₇	1	2	yes	—
332	C₂×C₄	#95×C₆₇	1	2	yes	—
333	C₂×C₄	#95×C₆₇	1	2	yes	—
334	C₂×C₄	#95×C₆₇	1	2	yes	—
335	C₂×C₄	#104×C₆₇	1	4	yes	—
336	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
337	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
338	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
339	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
340	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
341	C₂×C₄	#117×C₆₇	1	2	yes	—
342	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
343	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
344	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
345	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
346	C₂×C₄	#121×C₆₇	1	2	yes	—
347	C₂×C₄	#121×C₆₇	1	2	yes	—
348	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
349	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
350	C₂×C₄	#125×C₆₇	1	2	yes	—
351	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
352	C₈	—	2	8	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	2	no	—
360	C₂²	—	2	2	no	—
361	C₂²	C₂×#95×C₆₇	1	1	yes	characteristic
362	C₂²	#382×C₆₇	1	1	yes	characteristic
363	C₂²	#384×C₆₇	1	1	yes	characteristic
364	C₂²	#390×C₆₇	1	1	yes	characteristic
365	C₂²	#390×C₆₇	1	1	yes	characteristic
366	C₂²	#391×C₆₇	1	1	yes	characteristic
367	C₂²	#391×C₆₇	1	1	yes	characteristic
368	C₂²	C₂×#104×C₆₇	1	2	yes	—
369	C₂²	#421×C₆₇	1	2	yes	—
370	C₄	—	2	2	no	—
371	C₄	—	2	2	no	—
372	C₄	—	2	2	no	—
373	C₄	—	4	8	no	—
374	C₄	—	8	4	no	—
375	C₄	#339×C₆₇	1	4	yes	complemented
376	C₄	C₄×C₄⋊C₄×C₆₇	1	2	yes	—
377	C₄	#380×C₆₇	1	2	yes	—
378	C₄	#389×C₆₇	1	2	yes	—
379	C₄	#389×C₆₇	1	2	yes	—
380	C₂	—	2	1	no	—
381	C₂	—	2	1	no	—
382	C₂	—	2	1	no	—
383	C₂	—	2	1	no	—
384	C₂	C₆₇×C₄⋊#95	1	1	yes	characteristic
385	C₂	C₂×#339×C₆₇	1	1	yes	characteristic
386	C₂	C₆₇×#1733	1	1	yes	characteristic
387	C₂	C₆₇×#1980	1	2	yes	—
388	C₂	C₆₇×#1986	1	2	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. C₄×M₁₆×C₆₇	C₄	semidirect
14. #384×C₆₇	C₄	semidirect
15. #382×C₆₇	C₄	semidirect
21. C₄×C₄⋊C₄×C₆₇	C₄	semidirect
91. M₁₆×C₆₇	C₄⋊C₄	semidirect
180. C₄×C₆₇	#339	semidirect
185. #10851	C₆₇	direct (N × H)
199. C₆₇	#10851	direct (N × H)
206. C₄×M₁₆	C₄×C₆₇	semidirect
209. #384	C₄×C₆₇	semidirect
210. #382	C₄×C₆₇	semidirect
216. C₄×C₄⋊C₄	C₄×C₆₇	semidirect
286. M₁₆	C₄⋊C₄×C₆₇	semidirect
375. 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×C₄×M₁₆×C₆₇	2
C₆₇×#2285	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#339×C₆₇	2
#10851	67

Subgroups — C67×#10851 #30690102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10851 #30690102