Subgroups — C₄×#413×C₆₇ #30699647

Order 17152.

1570 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×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#413×C₆₇	C₂	1	2	yes	maximal
3	C₂×C₄×M₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₄×#101×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₄×#100×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1742	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	—
9	C₄×#41×C₆₇	—	2	2	no	—
10	C₄×#41×C₆₇	—	2	4	no	—
11	C₄×#41×C₆₇	—	2	2	no	—
12	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₆₇	C₄	1	4	yes	—
14	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
15	C₂×#117×C₆₇	C₄	1	2	yes	—
16	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₆₇	C₄	1	2	yes	—
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	#413×C₆₇	C₄	1	16	yes	direct product factor
20	C₂×#101×C₆₇	C₂²	1	4	yes	—
21	C₂×#100×C₆₇	C₂²	1	4	yes	—
22	C₄×M₁₆×C₆₇	—	2	4	no	—
23	C₄×M₁₆×C₆₇	C₂²	1	8	yes	—
24	C₂×#97×C₆₇	—	2	2	no	—
25	C₂×C₄×C₈×C₆₇	—	2	2	no	—
26	#343×C₆₇	C₂²	1	8	yes	—
27	C₂×#41×C₆₇	—	2	2	no	—
28	C₂×#41×C₆₇	—	2	2	no	—
29	C₂×#41×C₆₇	—	2	4	no	—
30	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
31	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
33	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
36	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₆₇	—	2	4	no	—
39	C₂×M₁₆×C₆₇	—	2	8	no	—
40	C₂×M₁₆×C₆₇	C₂×C₄	1	16	yes	complemented
41	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
42	C₂²×C₈×C₆₇	—	2	4	no	—
43	C₄×Q₈×C₆₇	—	2	4	no	—
44	C₄×Q₈×C₆₇	—	2	4	no	—
45	C₄×D₄×C₆₇	—	2	8	no	—
46	C₄×D₄×C₆₇	—	2	8	no	—
47	C₄×D₄×C₆₇	—	2	4	no	—
48	C₄×D₄×C₆₇	—	2	4	no	—
49	#117×C₆₇	—	2	2	no	—
50	#117×C₆₇	—	2	2	no	—
51	#117×C₆₇	—	2	4	no	—
52	#117×C₆₇	—	2	4	no	—
53	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
54	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
55	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
56	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₆₇	—	2	4	no	—
58	C₂×C₄²×C₆₇	—	2	2	no	—
59	C₂×C₄²×C₆₇	—	2	2	no	—
60	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
62	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	D₄	1	2	yes	—
65	C₂×C₄²×C₆₇	D₄	1	4	yes	—
66	C₂×C₄²×C₆₇	D₄	1	2	yes	—
67	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
68	#101×C₆₇	C₂×C₄	1	16	yes	complemented
69	#100×C₆₇	C₂×C₄	1	16	yes	complemented
70	#97×C₆₇	—	2	8	no	—
71	C₄×C₈×C₆₇	—	2	8	no	—
72	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
73	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	#41×C₆₇	—	2	16	no	—
75	#41×C₆₇	—	2	16	no	—
76	C₂×Q₈×C₆₇	—	2	2	no	—
77	C₂×Q₈×C₆₇	—	2	2	no	—
78	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
79	C₂×D₄×C₆₇	—	2	4	no	—
80	C₂×D₄×C₆₇	—	2	4	no	—
81	C₂×D₄×C₆₇	—	2	2	no	—
82	C₂×D₄×C₆₇	—	2	2	no	—
83	C₂×D₄×C₆₇	C₄²	1	8	yes	—
84	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	2	4	no	—
88	C₂²×C₄×C₆₇	—	2	4	no	—
89	C₂²×C₄×C₆₇	—	2	4	no	—
90	C₂²×C₄×C₆₇	—	4	4	no	—
91	C₂²×C₄×C₆₇	—	4	8	no	—
92	C₂²×C₄×C₆₇	C₄²	1	8	yes	—
93	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
94	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
103	C₂²×C₄×C₆₇	#41	1	4	yes	—
104	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
105	M₁₆×C₆₇	—	2	16	no	—
106	M₁₆×C₆₇	—	2	8	no	—
107	M₁₆×C₆₇	C₂²×C₄	1	16	yes	—
108	C₂×C₈×C₆₇	—	2	16	no	—
109	C₂×C₈×C₆₇	—	2	8	no	—
110	C₄⋊C₄×C₆₇	—	2	8	no	—
111	C₄⋊C₄×C₆₇	—	2	4	no	—
112	C₂²⋊C₄×C₆₇	—	2	4	no	—
113	C₂²⋊C₄×C₆₇	—	2	8	no	—
114	C₄²×C₆₇	—	2	2	no	—
115	C₄²×C₆₇	—	2	2	no	—
116	C₄²×C₆₇	—	2	4	no	—
117	C₄²×C₆₇	—	2	4	no	—
118	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
119	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
120	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
121	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
122	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
123	C₄²×C₆₇	C₂×D₄	1	4	yes	—
124	C₄²×C₆₇	C₂×D₄	1	4	yes	—
125	C₂³×C₆₇	—	2	2	no	—
126	C₂³×C₆₇	—	2	2	no	—
127	C₂³×C₆₇	—	4	2	no	—
128	C₂³×C₆₇	—	4	4	no	—
129	C₂³×C₆₇	C₂×C₄²	1	4	yes	—
130	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
131	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂³×C₆₇	#117	1	2	yes	—
133	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
134	Q₈×C₆₇	—	2	8	no	—
135	D₄×C₆₇	—	2	16	no	—
136	D₄×C₆₇	—	2	8	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	8	no	—
140	C₂×C₄×C₆₇	—	2	8	no	—
141	C₂×C₄×C₆₇	—	2	4	no	—
142	C₂×C₄×C₆₇	—	2	8	no	—
143	C₂×C₄×C₆₇	—	2	4	no	—
144	C₂×C₄×C₆₇	—	4	4	no	—
145	C₂×C₄×C₆₇	—	4	2	no	—
146	C₂×C₄×C₆₇	—	4	8	no	—
147	C₂×C₄×C₆₇	—	4	8	no	—
148	C₂×C₄×C₆₇	—	4	4	no	—
149	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
150	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
151	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
153	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
156	C₂×C₄×C₆₇	#117	1	2	yes	—
157	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
158	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
159	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
160	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
161	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
162	C₈×C₆₇	—	2	16	no	—
163	C₂²×C₆₇	—	2	2	no	—
164	C₂²×C₆₇	—	2	4	no	—
165	C₂²×C₆₇	—	2	4	no	—
166	C₂²×C₆₇	—	4	2	no	—
167	C₂²×C₆₇	—	4	2	no	—
168	C₂²×C₆₇	—	4	4	no	—
169	C₂²×C₆₇	—	4	4	no	—
170	C₂²×C₆₇	—	4	4	no	—
171	C₂²×C₆₇	C₄×C₂²⋊C₄	1	4	yes	—
172	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
173	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
174	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
175	C₄×C₆₇	—	2	2	no	—
176	C₄×C₆₇	—	2	4	no	—
177	C₄×C₆₇	—	2	4	no	—
178	C₄×C₆₇	—	4	4	no	—
179	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
180	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
181	C₄×C₆₇	#413	1	4	yes	direct product factor
182	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
183	C₄×#413	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂×C₆₇	—	2	1	no	—
185	C₂×C₆₇	—	2	1	no	—
186	C₂×C₆₇	—	2	4	no	—
187	C₂×C₆₇	—	4	4	no	—
188	C₂×C₆₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
189	C₂×C₆₇	C₂×#413	1	2	yes	—
190	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic
191	C₂×#413	C₂×C₆₇	1	2	yes	—
192	C₂×C₄×M₁₆	C₂×C₆₇	1	2	yes	complemented
193	C₄×#101	C₂×C₆₇	1	4	yes	complemented
194	C₄×#100	C₂×C₆₇	1	4	yes	complemented
195	#1742	C₂×C₆₇	1	2	yes	complemented
196	C₆₇	C₄×#413	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
198	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
199	C₄×#41	—	2	2	no	—
200	C₄×#41	—	2	2	no	—
201	C₄×#41	—	2	4	no	—
202	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
203	C₂×C₄×D₄	C₄×C₆₇	1	4	yes	—
204	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
205	C₂×#117	C₄×C₆₇	1	2	yes	—
206	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
207	C₂²×C₄²	C₄×C₆₇	1	2	yes	—
208	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
209	#413	C₄×C₆₇	1	16	yes	direct product factor
210	C₂×#101	C₂²×C₆₇	1	4	yes	—
211	C₂×#100	C₂²×C₆₇	1	4	yes	—
212	C₄×M₁₆	—	2	4	no	—
213	C₄×M₁₆	C₂²×C₆₇	1	8	yes	—
214	C₂×#97	—	2	2	no	—
215	C₂×C₄×C₈	—	2	2	no	—
216	#343	C₂²×C₆₇	1	8	yes	—
217	C₂×#41	—	2	2	no	—
218	C₂×#41	—	2	2	no	—
219	C₂×#41	—	2	4	no	—
220	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
221	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
222	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
223	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
224	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
225	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
226	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
227	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
228	C₂×M₁₆	—	2	4	no	—
229	C₂×M₁₆	—	2	8	no	—
230	C₂×M₁₆	C₂×C₄×C₆₇	1	16	yes	complemented
231	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
232	C₂²×C₈	—	2	4	no	—
233	C₄×Q₈	—	2	4	no	—
234	C₄×Q₈	—	2	4	no	—
235	C₄×D₄	—	2	8	no	—
236	C₄×D₄	—	2	8	no	—
237	C₄×D₄	—	2	4	no	—
238	C₄×D₄	—	2	4	no	—
239	#117	—	2	2	no	—
240	#117	—	2	2	no	—
241	#117	—	2	4	no	—
242	#117	—	2	4	no	—
243	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
244	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
245	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
246	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
247	C₂×C₄²	—	2	2	no	—
248	C₂×C₄²	—	2	4	no	—
249	C₂×C₄²	—	2	2	no	—
250	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
251	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
252	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
253	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
254	C₂×C₄²	D₄×C₆₇	1	2	yes	—
255	C₂×C₄²	D₄×C₆₇	1	2	yes	—
256	C₂×C₄²	D₄×C₆₇	1	4	yes	—
257	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
258	#101	C₂×C₄×C₆₇	1	16	yes	complemented
259	#100	C₂×C₄×C₆₇	1	16	yes	complemented
260	#97	—	2	8	no	—
261	C₄×C₈	—	2	8	no	—
262	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
263	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
264	#41	—	2	16	no	—
265	#41	—	2	16	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
269	C₂×D₄	—	2	4	no	—
270	C₂×D₄	—	2	4	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	C₄²×C₆₇	1	8	yes	—
274	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	2	4	no	—
278	C₂²×C₄	—	2	4	no	—
279	C₂²×C₄	—	2	4	no	—
280	C₂²×C₄	—	4	4	no	—
281	C₂²×C₄	—	4	8	no	—
282	C₂²×C₄	C₄²×C₆₇	1	8	yes	—
283	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
284	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
285	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
286	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
287	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
288	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
289	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
290	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
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	M₁₆	—	2	16	no	—
296	M₁₆	—	2	8	no	—
297	M₁₆	C₂²×C₄×C₆₇	1	16	yes	—
298	C₂×C₈	—	2	16	no	—
299	C₂×C₈	—	2	8	no	—
300	C₄⋊C₄	—	2	8	no	—
301	C₄⋊C₄	—	2	4	no	—
302	C₂²⋊C₄	—	2	4	no	—
303	C₂²⋊C₄	—	2	8	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	2	4	no	—
307	C₄²	—	2	4	no	—
308	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
309	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
310	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
311	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
312	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
313	C₄²	C₂×D₄×C₆₇	1	4	yes	—
314	C₄²	C₂×D₄×C₆₇	1	4	yes	—
315	C₂³	—	2	2	no	—
316	C₂³	—	2	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	4	4	no	—
319	C₂³	C₂×C₄²×C₆₇	1	4	yes	—
320	C₂³	C₂×C₄²×C₆₇	1	2	yes	—
321	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
322	C₂³	#117×C₆₇	1	2	yes	—
323	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
324	Q₈	—	2	8	no	—
325	D₄	—	2	16	no	—
326	D₄	—	2	8	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	8	no	—
330	C₂×C₄	—	2	8	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	8	no	—
333	C₂×C₄	—	2	4	no	—
334	C₂×C₄	—	4	4	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	8	no	—
337	C₂×C₄	—	4	8	no	—
338	C₂×C₄	—	4	4	no	—
339	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
340	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
341	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
342	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
343	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
344	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
345	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
346	C₂×C₄	#117×C₆₇	1	2	yes	—
347	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
348	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
349	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
350	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
351	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
352	C₈	—	2	16	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	2	4	no	—
355	C₂²	—	2	4	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	4	no	—
359	C₂²	—	4	4	no	—
360	C₂²	—	4	4	no	—
361	C₂²	C₄×C₂²⋊C₄×C₆₇	1	4	yes	—
362	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
363	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
364	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
365	C₄	—	2	4	no	—
366	C₄	—	2	2	no	—
367	C₄	—	2	4	no	—
368	C₄	—	4	4	no	—
369	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
370	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
371	C₄	#413×C₆₇	1	4	yes	direct product factor
372	C₄	C₂×C₄×D₄×C₆₇	1	4	yes	—
373	C₂	—	2	1	no	—
374	C₂	—	2	1	no	—
375	C₂	—	2	4	no	—
376	C₂	—	4	4	no	—
377	C₂	C₂×C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
378	C₂	C₂×#413×C₆₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
4. C₄×#101×C₆₇	C₂	semidirect
5. C₄×#100×C₆₇	C₂	semidirect
6. C₆₇×#1742	C₂	semidirect
19. #413×C₆₇	C₄	direct (N × H)
40. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
68. #101×C₆₇	C₂×C₄	semidirect
69. #100×C₆₇	C₂×C₄	semidirect
181. C₄×C₆₇	#413	direct (N × H)
183. C₄×#413	C₆₇	direct (N × H)
192. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
193. C₄×#101	C₂×C₆₇	semidirect
194. C₄×#100	C₂×C₆₇	semidirect
195. #1742	C₂×C₆₇	semidirect
196. C₆₇	C₄×#413	direct (N × H)
209. #413	C₄×C₆₇	direct (N × H)
230. C₂×M₁₆	C₂×C₄×C₆₇	semidirect
258. #101	C₂×C₄×C₆₇	semidirect
259. #100	C₂×C₄×C₆₇	semidirect
371. C₄	#413×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×#413×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₄×#101×C₆₇	2
C₄×#100×C₆₇	2
C₆₇×#1742	2
C₄×#413	67

Subgroups — C4×#413×C67 #30699647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#413×C₆₇ #30699647