Subgroups — C₆₇×#21982 #30701233

Order 17152.

966 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1947	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1852	C₂	1	4	yes	maximal
6	C₆₇×#1845	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1834	C₂	1	2	yes	maximal, complemented
8	C₂×#117×C₆₇	—	2	1	no	—
9	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
10	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
11	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
12	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	#431×C₆₇	C₄	1	8	yes	complemented
14	C₂×#107×C₆₇	C₂²	1	8	yes	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
18	#384×C₆₇	—	2	1	no	—
19	#384×C₆₇	—	2	1	no	—
20	#384×C₆₇	C₂²	1	2	yes	—
21	#384×C₆₇	C₂²	1	2	yes	—
22	#384×C₆₇	C₂²	1	1	yes	characteristic
23	#384×C₆₇	C₂²	1	1	yes	characteristic
24	#382×C₆₇	—	2	1	no	—
25	#380×C₆₇	—	2	1	no	—
26	#380×C₆₇	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
29	#340×C₆₇	C₂²	1	8	yes	—
30	C₂³×C₄×C₆₇	—	2	2	no	—
31	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
35	#117×C₆₇	—	4	4	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
46	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
47	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
48	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
51	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₄²×C₆₇	—	2	2	no	—
53	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
57	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	#107×C₆₇	—	4	8	no	—
60	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	complemented
61	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
62	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
63	#95×C₆₇	—	2	2	no	—
64	#95×C₆₇	—	2	2	no	—
65	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	—	2	2	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	4	no	—
73	C₂²×C₄×C₆₇	—	4	4	no	—
74	C₂²×C₄×C₆₇	—	4	4	no	—
75	C₂²×C₄×C₆₇	—	4	4	no	—
76	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
77	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
78	C₂²×C₄×C₆₇	D₈	1	4	yes	—
79	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
87	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
88	C₂×C₈×C₆₇	—	2	4	no	—
89	C₂×C₈×C₆₇	—	2	4	no	—
90	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
91	C₄⋊C₄×C₆₇	—	2	4	no	—
92	C₄⋊C₄×C₆₇	—	2	8	no	—
93	C₄⋊C₄×C₆₇	—	2	4	no	—
94	C₄⋊C₄×C₆₇	—	4	4	no	—
95	C₄⋊C₄×C₆₇	—	4	4	no	—
96	C₄⋊C₄×C₆₇	—	4	4	no	—
97	C₄⋊C₄×C₆₇	—	4	4	no	—
98	C₄⋊C₄×C₆₇	—	4	2	no	—
99	C₄⋊C₄×C₆₇	—	4	2	no	—
100	C₄⋊C₄×C₆₇	—	4	2	no	—
101	C₄⋊C₄×C₆₇	—	4	4	no	—
102	C₄⋊C₄×C₆₇	—	4	2	no	—
103	C₂²⋊C₄×C₆₇	—	4	4	no	—
104	C₄²×C₆₇	—	4	4	no	—
105	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
106	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
107	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
108	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
109	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
110	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
111	C₂³×C₆₇	—	2	1	no	—
112	C₂³×C₆₇	—	2	1	no	—
113	C₂³×C₆₇	—	2	1	no	—
114	C₂³×C₆₇	—	2	1	no	—
115	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂³×C₆₇	C₂×D₈	1	2	yes	—
117	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
118	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	2	4	no	—
124	C₂×C₄×C₆₇	—	2	4	no	—
125	C₂×C₄×C₆₇	—	2	8	no	—
126	C₂×C₄×C₆₇	—	4	4	no	—
127	C₂×C₄×C₆₇	—	4	2	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	4	4	no	—
132	C₂×C₄×C₆₇	—	4	4	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	—	8	8	no	—
136	C₂×C₄×C₆₇	#107	1	16	yes	complemented
137	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
145	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
146	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
147	C₂×C₄×C₆₇	#137	1	2	yes	—
148	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
150	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
151	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
152	C₈×C₆₇	—	2	8	no	—
153	C₂²×C₆₇	—	2	2	no	—
154	C₂²×C₆₇	—	2	2	no	—
155	C₂²×C₆₇	—	2	2	no	—
156	C₂²×C₆₇	—	2	2	no	—
157	C₂²×C₆₇	—	2	2	no	—
158	C₂²×C₆₇	—	2	2	no	—
159	C₂²×C₆₇	C₂×#107	1	4	yes	—
160	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₂²×C₆₇	#535	1	1	yes	characteristic
162	C₂²×C₆₇	#536	1	1	yes	characteristic
163	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
164	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
165	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
167	C₄×C₆₇	—	2	2	no	—
168	C₄×C₆₇	—	2	2	no	—
169	C₄×C₆₇	—	8	2	no	—
170	C₄×C₆₇	—	8	4	no	—
171	C₄×C₆₇	—	8	2	no	—
172	C₄×C₆₇	C₂×#107	1	4	yes	—
173	C₄×C₆₇	C₂×#107	1	4	yes	—
174	C₄×C₆₇	#428	1	4	yes	—
175	C₄×C₆₇	#520	1	2	yes	—
176	C₄×C₆₇	#544	1	2	yes	—
177	#21982	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂×C₆₇	—	2	4	no	—
179	C₂×C₆₇	#2317	1	1	yes	characteristic
180	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
181	C₂×C₆₇	C₂×#428	1	1	yes	characteristic
182	C₂×C₆₇	#3205	1	1	yes	characteristic
183	C₂×C₆₇	#3209	1	1	yes	characteristic
184	C₂×C₆₇	#3226	1	1	yes	characteristic
185	C₂×C₆₇	#3230	1	1	yes	characteristic
186	C₂×#431	C₂×C₆₇	1	1	yes	characteristic
187	#2285	C₂×C₆₇	1	1	yes	characteristic
188	C₂×#384	C₂×C₆₇	1	1	yes	characteristic
189	#1947	C₂×C₆₇	1	4	yes	complemented
190	#1852	C₂×C₆₇	1	4	yes	—
191	#1845	C₂×C₆₇	1	2	yes	complemented
192	#1834	C₂×C₆₇	1	2	yes	complemented
193	C₆₇	#21982	1	1	yes	characteristic, direct product factor, Sylow
194	C₂×#117	—	2	1	no	—
195	C₂²×C₄⋊C₄	—	2	1	no	—
196	C₂²×C₄⋊C₄	—	2	1	no	—
197	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
198	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
199	#431	C₄×C₆₇	1	8	yes	complemented
200	C₂×#107	C₂²×C₆₇	1	8	yes	—
201	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
202	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
203	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
204	#384	—	2	1	no	—
205	#384	—	2	1	no	—
206	#384	C₂²×C₆₇	1	1	yes	characteristic
207	#384	C₂²×C₆₇	1	2	yes	—
208	#384	C₂²×C₆₇	1	1	yes	characteristic
209	#384	C₂²×C₆₇	1	2	yes	—
210	#382	—	2	1	no	—
211	#380	—	2	1	no	—
212	#380	C₂²×C₆₇	1	2	yes	—
213	C₄×C₄⋊C₄	—	2	1	no	—
214	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
215	#340	C₂²×C₆₇	1	8	yes	—
216	C₂³×C₄	—	2	2	no	—
217	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
218	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
219	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
220	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
221	#117	—	4	4	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	—	4	4	no	—
232	C₂×C₄⋊C₄	—	4	4	no	—
233	C₂×C₄⋊C₄	—	4	4	no	—
234	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
235	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
236	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
237	C₂×C₂²⋊C₄	—	2	2	no	—
238	C₂×C₄²	—	2	2	no	—
239	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
240	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
241	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
242	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
243	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
244	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
245	#107	—	4	8	no	—
246	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	complemented
247	C₂²⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
248	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
249	#95	—	2	2	no	—
250	#95	—	2	2	no	—
251	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	4	4	no	—
260	C₂²×C₄	—	4	4	no	—
261	C₂²×C₄	—	4	4	no	—
262	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
263	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
264	C₂²×C₄	D₈×C₆₇	1	4	yes	—
265	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	—
266	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
267	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×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×Q₈×C₆₇	1	2	yes	—
273	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	2	8	no	—
279	C₄⋊C₄	—	2	4	no	—
280	C₄⋊C₄	—	4	4	no	—
281	C₄⋊C₄	—	4	4	no	—
282	C₄⋊C₄	—	4	4	no	—
283	C₄⋊C₄	—	4	4	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	4	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	4	no	—
290	C₄²	—	4	4	no	—
291	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
292	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
293	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
294	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
295	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
296	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	2	1	no	—
301	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
302	C₂³	C₂×D₈×C₆₇	1	2	yes	—
303	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
304	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	8	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	8	8	no	—
322	C₂×C₄	#107×C₆₇	1	16	yes	complemented
323	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
324	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
325	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
326	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
327	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
328	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
329	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
330	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
331	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
332	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
333	C₂×C₄	#137×C₆₇	1	2	yes	—
334	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
335	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
336	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
337	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
338	C₈	—	2	8	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	C₂×#107×C₆₇	1	4	yes	—
346	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
347	C₂²	#535×C₆₇	1	1	yes	characteristic
348	C₂²	#536×C₆₇	1	1	yes	characteristic
349	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
350	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
351	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
352	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	8	4	no	—
356	C₄	—	8	2	no	—
357	C₄	—	8	2	no	—
358	C₄	C₂×#107×C₆₇	1	4	yes	—
359	C₄	C₂×#107×C₆₇	1	4	yes	—
360	C₄	#428×C₆₇	1	4	yes	—
361	C₄	#520×C₆₇	1	2	yes	—
362	C₄	#544×C₆₇	1	2	yes	—
363	C₂	—	2	4	no	—
364	C₂	C₆₇×#2317	1	1	yes	characteristic
365	C₂	C₂²×#107×C₆₇	1	1	yes	characteristic
366	C₂	C₂×#428×C₆₇	1	1	yes	characteristic
367	C₂	C₆₇×#3205	1	1	yes	characteristic
368	C₂	C₆₇×#3209	1	1	yes	characteristic
369	C₂	C₆₇×#3226	1	1	yes	characteristic
370	C₂	C₆₇×#3230	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1947	C₂	semidirect
6. C₆₇×#1845	C₂	semidirect
7. C₆₇×#1834	C₂	semidirect
13. #431×C₆₇	C₄	semidirect
60. C₄⋊C₈×C₆₇	C₂×C₄	semidirect
62. C₄×C₈×C₆₇	C₂×C₄	semidirect
136. C₂×C₄×C₆₇	#107	semidirect
177. #21982	C₆₇	direct (N × H)
189. #1947	C₂×C₆₇	semidirect
191. #1845	C₂×C₆₇	semidirect
192. #1834	C₂×C₆₇	semidirect
193. C₆₇	#21982	direct (N × H)
199. #431	C₄×C₆₇	semidirect
246. C₄⋊C₈	C₂×C₄×C₆₇	semidirect
248. C₄×C₈	C₂×C₄×C₆₇	semidirect
322. C₂×C₄	#107×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431×C₆₇	2
C₆₇×#2285	2
C₂×#384×C₆₇	2
C₆₇×#1947	2
C₆₇×#1852	2
C₆₇×#1845	2
C₆₇×#1834	2
#21982	67

Subgroups — C67×#21982 #30701233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21982 #30701233