Subgroups — C₄×#416×C₆₇ #30699659

Order 17152.

1670 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#416×C₆₇	C₂	1	2	yes	maximal
3	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#103×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₄×#102×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1735	C₂	1	2	yes	maximal, complemented
8	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
10	C₄×#41×C₆₇	—	2	4	no	—
11	C₄×#41×C₆₇	C₄	1	8	yes	complemented
12	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₆₇	—	2	2	no	—
14	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
15	C₂×#117×C₆₇	—	2	1	no	—
16	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#117×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₆₇	—	2	1	no	—
19	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	#416×C₆₇	C₄	1	16	yes	direct product factor
21	C₂×#103×C₆₇	C₂²	1	4	yes	—
22	C₂×#102×C₆₇	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₆₇	—	2	4	no	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
25	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
26	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
27	#376×C₆₇	—	2	1	no	—
28	C₄³×C₆₇	—	2	1	no	—
29	#340×C₆₇	C₂²	1	8	yes	—
30	C₂×#41×C₆₇	—	2	2	no	—
31	C₂×#41×C₆₇	—	2	4	no	—
32	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
33	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
34	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₂²×D₄×C₆₇	—	2	1	no	—
36	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	—	2	1	no	—
38	C₂³×C₄×C₆₇	—	2	2	no	—
39	C₂³×C₄×C₆₇	D₄	1	2	yes	—
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	—	2	4	no	—
42	C₂²×C₈×C₆₇	—	2	4	no	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
44	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
45	C₄×Q₈×C₆₇	—	2	4	no	—
46	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
47	C₄×D₄×C₆₇	—	2	16	no	—
48	C₄×D₄×C₆₇	—	2	4	no	—
49	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
50	#117×C₆₇	—	2	2	no	—
51	#117×C₆₇	—	2	4	no	—
52	#117×C₆₇	—	2	8	no	—
53	#117×C₆₇	C₂×C₄	1	4	yes	—
54	#117×C₆₇	C₂×C₄	1	8	yes	complemented
55	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
58	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄²×C₆₇	—	2	1	no	—
62	C₂×C₄²×C₆₇	—	2	2	no	—
63	C₂×C₄²×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	—	2	8	no	—
65	C₂×C₄²×C₆₇	—	2	4	no	—
66	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
67	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	D₄	1	4	yes	—
69	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
71	#103×C₆₇	C₂×C₄	1	16	yes	complemented
72	#102×C₆₇	C₂×C₄	1	16	yes	complemented
73	C₄×C₈×C₆₇	—	2	8	no	—
74	#95×C₆₇	—	2	4	no	—
75	C₂⁴×C₆₇	—	2	1	no	—
76	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
77	#41×C₆₇	—	2	8	no	—
78	#41×C₆₇	—	2	16	no	—
79	#41×C₆₇	C₄²	1	16	yes	complemented
80	C₂×Q₈×C₆₇	—	2	1	no	—
81	C₂×Q₈×C₆₇	—	2	2	no	—
82	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
83	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂×D₄×C₆₇	—	2	4	no	—
85	C₂×D₄×C₆₇	—	2	8	no	—
86	C₂×D₄×C₆₇	—	2	1	no	—
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
89	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₆₇	—	2	1	no	—
91	C₂²×C₄×C₆₇	—	2	4	no	—
92	C₂²×C₄×C₆₇	—	2	8	no	—
93	C₂²×C₄×C₆₇	—	2	8	no	—
94	C₂²×C₄×C₆₇	—	2	4	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	2	8	no	—
97	C₂²×C₄×C₆₇	—	4	8	no	—
98	C₂²×C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
99	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₆₇	#41	1	4	yes	—
106	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
107	C₂×C₈×C₆₇	—	2	16	no	—
108	C₂×C₈×C₆₇	—	2	4	no	—
109	C₂×C₈×C₆₇	—	2	4	no	—
110	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
111	C₄⋊C₄×C₆₇	—	2	8	no	—
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₄⋊C₄×C₆₇	—	2	4	no	—
114	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
115	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
116	C₂²⋊C₄×C₆₇	—	2	4	no	—
117	C₂²⋊C₄×C₆₇	—	2	8	no	—
118	C₂²⋊C₄×C₆₇	—	2	8	no	—
119	C₄²×C₆₇	—	2	4	no	—
120	C₄²×C₆₇	—	2	8	no	—
121	C₄²×C₆₇	—	2	8	no	—
122	C₄²×C₆₇	—	4	8	no	—
123	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
124	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
125	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
126	C₄²×C₆₇	C₂×D₄	1	4	yes	—
127	C₂³×C₆₇	—	2	4	no	—
128	C₂³×C₆₇	—	2	2	no	—
129	C₂³×C₆₇	—	2	4	no	—
130	C₂³×C₆₇	—	4	4	no	—
131	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂³×C₆₇	C₄×D₄	1	4	yes	—
133	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
134	Q₈×C₆₇	—	2	2	no	—
135	Q₈×C₆₇	—	2	4	no	—
136	Q₈×C₆₇	C₂×C₄²	1	4	yes	—
137	D₄×C₆₇	—	2	16	no	—
138	D₄×C₆₇	—	2	2	no	—
139	D₄×C₆₇	—	2	4	no	—
140	D₄×C₆₇	C₂×C₄²	1	4	yes	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	4	no	—
143	C₂×C₄×C₆₇	—	2	4	no	—
144	C₂×C₄×C₆₇	—	2	16	no	—
145	C₂×C₄×C₆₇	—	2	16	no	—
146	C₂×C₄×C₆₇	—	2	4	no	—
147	C₂×C₄×C₆₇	—	2	8	no	—
148	C₂×C₄×C₆₇	—	4	8	no	—
149	C₂×C₄×C₆₇	—	4	4	no	—
150	C₂×C₄×C₆₇	—	4	16	no	—
151	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
152	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
153	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
154	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
155	C₂×C₄×C₆₇	#117	1	4	yes	—
156	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
157	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
158	C₂×C₄×C₆₇	#135	1	8	yes	—
159	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
161	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
163	C₈×C₆₇	—	2	8	no	—
164	C₂²×C₆₇	—	2	4	no	—
165	C₂²×C₆₇	—	2	8	no	—
166	C₂²×C₆₇	—	2	4	no	—
167	C₂²×C₆₇	—	4	4	no	—
168	C₂²×C₆₇	—	4	8	no	—
169	C₂²×C₆₇	C₄×C₂²⋊C₄	1	4	yes	—
170	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
171	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
172	C₂²×C₆₇	C₂×#135	1	4	yes	—
173	C₄×C₆₇	—	2	4	no	—
174	C₄×C₆₇	—	2	4	no	—
175	C₄×C₆₇	—	4	4	no	—
176	C₄×C₆₇	—	4	8	no	—
177	C₄×C₆₇	C₄×C₂²⋊C₄	1	4	yes	—
178	C₄×C₆₇	#416	1	8	yes	direct product factor
179	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
180	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
181	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
182	C₄×#416	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂×C₆₇	—	2	4	no	—
184	C₂×C₆₇	—	4	4	no	—
185	C₂×C₆₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₆₇	C₂×#416	1	2	yes	—
187	C₂×C₆₇	C₄×#135	1	4	yes	—
188	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic
189	C₂×#416	C₂×C₆₇	1	2	yes	—
190	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
191	C₄×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
192	C₄×#103	C₂×C₆₇	1	4	yes	complemented
193	C₄×#102	C₂×C₆₇	1	4	yes	complemented
194	#1735	C₂×C₆₇	1	2	yes	complemented
195	C₆₇	C₄×#416	1	1	yes	characteristic, direct product factor, Sylow
196	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
197	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
198	C₄×#41	—	2	4	no	—
199	C₄×#41	C₄×C₆₇	1	8	yes	complemented
200	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
201	C₂×C₄×D₄	—	2	2	no	—
202	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
203	C₂×#117	—	2	1	no	—
204	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
205	C₂×#117	C₂²×C₆₇	1	2	yes	—
206	C₂²×C₄²	—	2	1	no	—
207	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
208	#416	C₄×C₆₇	1	16	yes	direct product factor
209	C₂×#103	C₂²×C₆₇	1	4	yes	—
210	C₂×#102	C₂²×C₆₇	1	4	yes	—
211	C₂×C₄×C₈	—	2	4	no	—
212	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
213	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
214	C₄×C₂²⋊C₄	—	2	2	no	—
215	#376	—	2	1	no	—
216	C₄³	—	2	1	no	—
217	#340	C₂²×C₆₇	1	8	yes	—
218	C₂×#41	—	2	2	no	—
219	C₂×#41	—	2	4	no	—
220	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
221	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
222	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
223	C₂²×D₄	—	2	1	no	—
224	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
225	C₂³×C₄	—	2	1	no	—
226	C₂³×C₄	—	2	2	no	—
227	C₂³×C₄	D₄×C₆₇	1	2	yes	—
228	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
229	C₂²×C₈	—	2	4	no	—
230	C₂²×C₈	—	2	4	no	—
231	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
232	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
233	C₄×Q₈	—	2	4	no	—
234	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	—
235	C₄×D₄	—	2	16	no	—
236	C₄×D₄	—	2	4	no	—
237	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
238	#117	—	2	2	no	—
239	#117	—	2	4	no	—
240	#117	—	2	8	no	—
241	#117	C₂×C₄×C₆₇	1	4	yes	—
242	#117	C₂×C₄×C₆₇	1	8	yes	complemented
243	C₂×C₄⋊C₄	—	2	2	no	—
244	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
245	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
246	C₂×C₂²⋊C₄	—	2	1	no	—
247	C₂×C₂²⋊C₄	—	2	2	no	—
248	C₂×C₂²⋊C₄	—	2	2	no	—
249	C₂×C₄²	—	2	1	no	—
250	C₂×C₄²	—	2	2	no	—
251	C₂×C₄²	—	2	2	no	—
252	C₂×C₄²	—	2	8	no	—
253	C₂×C₄²	—	2	4	no	—
254	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
255	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
256	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
257	C₂×C₄²	D₄×C₆₇	1	4	yes	—
258	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
259	#103	C₂×C₄×C₆₇	1	16	yes	complemented
260	#102	C₂×C₄×C₆₇	1	16	yes	complemented
261	C₄×C₈	—	2	8	no	—
262	#95	—	2	4	no	—
263	C₂⁴	—	2	1	no	—
264	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
265	#41	—	2	8	no	—
266	#41	—	2	16	no	—
267	#41	C₄²×C₆₇	1	16	yes	complemented
268	C₂×Q₈	—	2	1	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
271	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
272	C₂×D₄	—	2	4	no	—
273	C₂×D₄	—	2	8	no	—
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
277	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
278	C₂²×C₄	—	2	1	no	—
279	C₂²×C₄	—	2	4	no	—
280	C₂²×C₄	—	2	8	no	—
281	C₂²×C₄	—	2	8	no	—
282	C₂²×C₄	—	2	4	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	8	no	—
285	C₂²×C₄	—	4	8	no	—
286	C₂²×C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
287	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
288	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
289	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
290	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
291	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
292	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
293	C₂²×C₄	#41×C₆₇	1	4	yes	—
294	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
295	C₂×C₈	—	2	16	no	—
296	C₂×C₈	—	2	4	no	—
297	C₂×C₈	—	2	4	no	—
298	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
299	C₄⋊C₄	—	2	8	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	4	no	—
302	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
303	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
304	C₂²⋊C₄	—	2	4	no	—
305	C₂²⋊C₄	—	2	8	no	—
306	C₂²⋊C₄	—	2	8	no	—
307	C₄²	—	2	4	no	—
308	C₄²	—	2	8	no	—
309	C₄²	—	2	8	no	—
310	C₄²	—	4	8	no	—
311	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
312	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
313	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
314	C₄²	C₂×D₄×C₆₇	1	4	yes	—
315	C₂³	—	2	4	no	—
316	C₂³	—	2	2	no	—
317	C₂³	—	2	4	no	—
318	C₂³	—	4	4	no	—
319	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
320	C₂³	C₄×D₄×C₆₇	1	4	yes	—
321	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
322	Q₈	—	2	2	no	—
323	Q₈	—	2	4	no	—
324	Q₈	C₂×C₄²×C₆₇	1	4	yes	—
325	D₄	—	2	16	no	—
326	D₄	—	2	2	no	—
327	D₄	—	2	4	no	—
328	D₄	C₂×C₄²×C₆₇	1	4	yes	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	16	no	—
333	C₂×C₄	—	2	16	no	—
334	C₂×C₄	—	2	4	no	—
335	C₂×C₄	—	2	8	no	—
336	C₂×C₄	—	4	8	no	—
337	C₂×C₄	—	4	4	no	—
338	C₂×C₄	—	4	16	no	—
339	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
340	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
341	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
342	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
343	C₂×C₄	#117×C₆₇	1	4	yes	—
344	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
345	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
346	C₂×C₄	#135×C₆₇	1	8	yes	—
347	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
349	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
350	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
351	C₈	—	2	8	no	—
352	C₂²	—	2	4	no	—
353	C₂²	—	2	8	no	—
354	C₂²	—	2	4	no	—
355	C₂²	—	4	4	no	—
356	C₂²	—	4	8	no	—
357	C₂²	C₄×C₂²⋊C₄×C₆₇	1	4	yes	—
358	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
359	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
360	C₂²	C₂×#135×C₆₇	1	4	yes	—
361	C₄	—	2	4	no	—
362	C₄	—	2	4	no	—
363	C₄	—	4	4	no	—
364	C₄	—	4	8	no	—
365	C₄	C₄×C₂²⋊C₄×C₆₇	1	4	yes	—
366	C₄	#416×C₆₇	1	8	yes	direct product factor
367	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
368	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
369	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
370	C₂	—	2	4	no	—
371	C₂	—	4	4	no	—
372	C₂	C₂×C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
373	C₂	C₂×#416×C₆₇	1	2	yes	—
374	C₂	C₄×#135×C₆₇	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₆₇	D₄
commutator	C₄	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄×C₈×C₆₇	C₂	semidirect
4. C₄×#117×C₆₇	C₂	semidirect
5. C₄×#103×C₆₇	C₂	semidirect
6. C₄×#102×C₆₇	C₂	semidirect
7. C₆₇×#1735	C₂	semidirect
11. C₄×#41×C₆₇	C₄	semidirect
20. #416×C₆₇	C₄	direct (N × H)
43. C₂²×C₈×C₆₇	C₂×C₄	semidirect
54. #117×C₆₇	C₂×C₄	semidirect
71. #103×C₆₇	C₂×C₄	semidirect
72. #102×C₆₇	C₂×C₄	semidirect
79. #41×C₆₇	C₄²	semidirect
178. C₄×C₆₇	#416	direct (N × H)
182. C₄×#416	C₆₇	direct (N × H)
190. C₂²×C₄×C₈	C₂×C₆₇	semidirect
191. C₄×#117	C₂×C₆₇	semidirect
192. C₄×#103	C₂×C₆₇	semidirect
193. C₄×#102	C₂×C₆₇	semidirect
194. #1735	C₂×C₆₇	semidirect
195. C₆₇	C₄×#416	direct (N × H)
199. C₄×#41	C₄×C₆₇	semidirect
208. #416	C₄×C₆₇	direct (N × H)
231. C₂²×C₈	C₂×C₄×C₆₇	semidirect
242. #117	C₂×C₄×C₆₇	semidirect
259. #103	C₂×C₄×C₆₇	semidirect
260. #102	C₂×C₄×C₆₇	semidirect
267. #41	C₄²×C₆₇	semidirect
366. C₄	#416×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₆₇	2
C₂×#416×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₄×#117×C₆₇	2
C₄×#103×C₆₇	2
C₄×#102×C₆₇	2
C₆₇×#1735	2
C₄×#416	67

Subgroups — C4×#416×C67 #30699659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#416×C₆₇ #30699659