Subgroups — C₂×C₄×#104×C₆₇ #30699653

Order 17152.

1878 conjugacy classes of subgroups.

The subgroup lattice has 452 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	C₂²×#104×C₆₇	C₂	1	2	yes	maximal
3	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#104×C₆₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×#339×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	C₄×#41×C₆₇	—	2	2	no	—
10	C₄×#41×C₆₇	—	2	2	no	—
11	C₄×#41×C₆₇	C₄	1	4	yes	complemented
12	C₄×#41×C₆₇	C₂²	1	4	yes	—
13	C₂×C₄×Q₈×C₆₇	C₄	1	2	yes	complemented
14	C₂×C₄×D₄×C₆₇	—	2	2	no	—
15	C₂×C₄×D₄×C₆₇	C₄	1	2	yes	complemented
16	C₂×#117×C₆₇	—	2	1	no	—
17	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₆₇	—	2	2	no	—
19	C₂²×C₄²×C₆₇	—	2	1	no	—
20	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#104×C₆₇	C₄	1	16	yes	direct product factor
23	C₂×#104×C₆₇	C₂²	1	8	yes	—
24	C₄×M₁₆×C₆₇	—	2	2	no	—
25	C₄×M₁₆×C₆₇	C₂²	1	4	yes	complemented
26	C₂×#97×C₆₇	—	2	1	no	—
27	C₂×C₄×C₈×C₆₇	—	2	1	no	—
28	C₄³×C₆₇	—	2	2	no	—
29	C₄³×C₆₇	C₂²	1	4	yes	complemented
30	#339×C₆₇	C₂²	1	8	yes	complemented
31	C₂×#41×C₆₇	—	2	1	no	—
32	C₂×#41×C₆₇	—	2	1	no	—
33	C₂×#41×C₆₇	—	2	4	no	—
34	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
35	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×#41×C₆₇	C₂³	1	2	yes	—
37	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₄×C₆₇	—	2	1	no	—
39	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	—	2	2	no	—
41	C₂³×C₄×C₆₇	—	2	1	no	—
42	C₂³×C₄×C₆₇	—	2	2	no	—
43	C₂³×C₄×C₆₇	D₄	1	2	yes	—
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₆₇	—	2	2	no	—
46	C₂×M₁₆×C₆₇	—	2	4	no	—
47	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
48	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₄×Q₈×C₆₇	—	2	4	no	—
51	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
52	C₄×D₄×C₆₇	—	2	8	no	—
53	C₄×D₄×C₆₇	—	2	8	no	—
54	C₄×D₄×C₆₇	—	2	4	no	—
55	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
56	#117×C₆₇	—	2	1	no	—
57	#117×C₆₇	—	2	1	no	—
58	#117×C₆₇	—	2	4	no	—
59	#117×C₆₇	—	2	4	no	—
60	#117×C₆₇	C₂×C₄	1	2	yes	—
61	#117×C₆₇	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₄²×C₆₇	—	2	16	no	—
67	C₂×C₄²×C₆₇	—	2	2	no	—
68	C₂×C₄²×C₆₇	—	2	1	no	—
69	C₂×C₄²×C₆₇	—	2	8	no	—
70	C₂×C₄²×C₆₇	—	2	2	no	—
71	C₂×C₄²×C₆₇	—	2	4	no	—
72	C₂×C₄²×C₆₇	—	2	4	no	—
73	C₂×C₄²×C₆₇	—	2	4	no	—
74	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
75	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
76	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	complemented
77	C₂×C₄²×C₆₇	D₄	1	2	yes	—
78	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	D₄	1	2	yes	—
80	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
82	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
83	#104×C₆₇	C₂×C₄	1	32	yes	direct product factor
84	#97×C₆₇	—	2	4	no	—
85	C₄×C₈×C₆₇	—	2	4	no	—
86	C₂⁴×C₆₇	—	2	1	no	—
87	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
88	#41×C₆₇	—	2	4	no	—
89	#41×C₆₇	—	2	8	no	—
90	#41×C₆₇	—	2	8	no	—
91	#41×C₆₇	—	2	4	no	—
92	#41×C₆₇	C₄²	1	8	yes	complemented
93	#41×C₆₇	C₂²×C₄	1	8	yes	—
94	C₂×Q₈×C₆₇	—	2	1	no	—
95	C₂×Q₈×C₆₇	—	2	2	no	—
96	C₂×Q₈×C₆₇	C₄²	1	4	yes	complemented
97	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	2	8	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	2	1	no	—
102	C₂×D₄×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	C₄²	1	4	yes	complemented
104	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₆₇	—	2	16	no	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	8	no	—
108	C₂²×C₄×C₆₇	—	2	4	no	—
109	C₂²×C₄×C₆₇	—	2	8	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	8	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	8	no	—
115	C₂²×C₄×C₆₇	—	4	8	no	—
116	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
118	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
122	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
123	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	#41	1	4	yes	—
127	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
128	M₁₆×C₆₇	—	2	8	no	—
129	M₁₆×C₆₇	—	2	4	no	—
130	M₁₆×C₆₇	C₂²×C₄	1	8	yes	complemented
131	C₂×C₈×C₆₇	—	2	8	no	—
132	C₂×C₈×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	—	2	8	no	—
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
136	C₂²⋊C₄×C₆₇	—	2	2	no	—
137	C₂²⋊C₄×C₆₇	—	2	2	no	—
138	C₂²⋊C₄×C₆₇	—	2	8	no	—
139	C₄²×C₆₇	—	2	32	no	—
140	C₄²×C₆₇	—	2	2	no	—
141	C₄²×C₆₇	—	2	8	no	—
142	C₄²×C₆₇	—	2	8	no	—
143	C₄²×C₆₇	—	2	2	no	—
144	C₄²×C₆₇	—	2	4	no	—
145	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
146	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
147	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
148	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
149	C₄²×C₆₇	C₂²×C₄	1	8	yes	complemented
150	C₄²×C₆₇	C₂×D₄	1	2	yes	—
151	C₄²×C₆₇	C₂×D₄	1	2	yes	—
152	C₂³×C₆₇	—	2	4	no	—
153	C₂³×C₆₇	—	2	2	no	—
154	C₂³×C₆₇	—	2	4	no	—
155	C₂³×C₆₇	—	4	4	no	—
156	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂³×C₆₇	C₄×D₄	1	4	yes	—
159	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
160	Q₈×C₆₇	—	2	2	no	—
161	Q₈×C₆₇	—	2	4	no	—
162	Q₈×C₆₇	C₂×C₄²	1	4	yes	complemented
163	D₄×C₆₇	—	2	8	no	—
164	D₄×C₆₇	—	2	8	no	—
165	D₄×C₆₇	—	2	2	no	—
166	D₄×C₆₇	—	2	4	no	—
167	D₄×C₆₇	C₂×C₄²	1	4	yes	complemented
168	C₂×C₄×C₆₇	—	2	16	no	—
169	C₂×C₄×C₆₇	—	2	32	no	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	8	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	4	no	—
174	C₂×C₄×C₆₇	—	2	8	no	—
175	C₂×C₄×C₆₇	—	2	8	no	—
176	C₂×C₄×C₆₇	—	2	8	no	—
177	C₂×C₄×C₆₇	—	2	8	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	4	4	no	—
180	C₂×C₄×C₆₇	—	4	16	no	—
181	C₂×C₄×C₆₇	#104	1	8	yes	direct product factor
182	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
183	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
184	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
185	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
188	C₂×C₄×C₆₇	#117	1	2	yes	—
189	C₂×C₄×C₆₇	#117	1	2	yes	—
190	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
191	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
192	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
193	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
195	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
197	C₈×C₆₇	—	2	8	no	—
198	C₂²×C₆₇	—	2	2	no	—
199	C₂²×C₆₇	—	2	2	no	—
200	C₂²×C₆₇	—	2	8	no	—
201	C₂²×C₆₇	—	2	4	no	—
202	C₂²×C₆₇	—	4	4	no	—
203	C₂²×C₆₇	—	4	8	no	—
204	C₂²×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
205	C₂²×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
206	C₂²×C₆₇	C₂×#104	1	4	yes	—
207	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
208	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
209	C₄×C₆₇	—	2	16	no	—
210	C₄×C₆₇	—	2	4	no	—
211	C₄×C₆₇	—	2	4	no	—
212	C₄×C₆₇	—	4	4	no	—
213	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
214	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
215	C₄×C₆₇	C₂×#104	1	8	yes	direct product factor
216	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
217	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
218	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
219	C₂×C₄×#104	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂×C₆₇	—	2	2	no	—
221	C₂×C₆₇	—	2	2	no	—
222	C₂×C₆₇	—	4	4	no	—
223	C₂×C₆₇	C₄×#104	1	4	yes	direct product factor
224	C₂×C₆₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
225	C₂×C₆₇	C₂²×#104	1	2	yes	—
226	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic
227	C₂²×#104	C₂×C₆₇	1	2	yes	—
228	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
229	C₂×C₄³	C₂×C₆₇	1	1	yes	characteristic, complemented
230	C₄×#104	C₂×C₆₇	1	8	yes	direct product factor
231	C₂×#339	C₂×C₆₇	1	2	yes	complemented
232	C₆₇	C₂×C₄×#104	1	1	yes	characteristic, direct product factor, Sylow
233	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
234	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
235	C₄×#41	—	2	2	no	—
236	C₄×#41	—	2	2	no	—
237	C₄×#41	C₄×C₆₇	1	4	yes	complemented
238	C₄×#41	C₂²×C₆₇	1	4	yes	—
239	C₂×C₄×Q₈	C₄×C₆₇	1	2	yes	complemented
240	C₂×C₄×D₄	—	2	2	no	—
241	C₂×C₄×D₄	C₄×C₆₇	1	2	yes	complemented
242	C₂×#117	—	2	1	no	—
243	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
244	C₂²×C₄²	—	2	2	no	—
245	C₂²×C₄²	—	2	1	no	—
246	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
247	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
248	C₂×#104	C₄×C₆₇	1	16	yes	direct product factor
249	C₂×#104	C₂²×C₆₇	1	8	yes	—
250	C₄×M₁₆	—	2	2	no	—
251	C₄×M₁₆	C₂²×C₆₇	1	4	yes	complemented
252	C₂×#97	—	2	1	no	—
253	C₂×C₄×C₈	—	2	1	no	—
254	C₄³	—	2	2	no	—
255	C₄³	C₂²×C₆₇	1	4	yes	complemented
256	#339	C₂²×C₆₇	1	8	yes	complemented
257	C₂×#41	—	2	1	no	—
258	C₂×#41	—	2	1	no	—
259	C₂×#41	—	2	4	no	—
260	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
261	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
262	C₂×#41	C₂³×C₆₇	1	2	yes	—
263	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
264	C₂²×D₄	—	2	1	no	—
265	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
266	C₂³×C₄	—	2	2	no	—
267	C₂³×C₄	—	2	1	no	—
268	C₂³×C₄	—	2	2	no	—
269	C₂³×C₄	D₄×C₆₇	1	2	yes	—
270	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
271	C₂×M₁₆	—	2	2	no	—
272	C₂×M₁₆	—	2	4	no	—
273	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
274	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
275	C₂²×C₈	—	2	2	no	—
276	C₄×Q₈	—	2	4	no	—
277	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
278	C₄×D₄	—	2	8	no	—
279	C₄×D₄	—	2	8	no	—
280	C₄×D₄	—	2	4	no	—
281	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
282	#117	—	2	1	no	—
283	#117	—	2	1	no	—
284	#117	—	2	4	no	—
285	#117	—	2	4	no	—
286	#117	C₂×C₄×C₆₇	1	2	yes	—
287	#117	C₂³×C₆₇	1	2	yes	—
288	C₂×C₄⋊C₄	—	2	2	no	—
289	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
290	C₂×C₂²⋊C₄	—	2	1	no	—
291	C₂×C₂²⋊C₄	—	2	2	no	—
292	C₂×C₄²	—	2	16	no	—
293	C₂×C₄²	—	2	2	no	—
294	C₂×C₄²	—	2	1	no	—
295	C₂×C₄²	—	2	2	no	—
296	C₂×C₄²	—	2	8	no	—
297	C₂×C₄²	—	2	4	no	—
298	C₂×C₄²	—	2	4	no	—
299	C₂×C₄²	—	2	4	no	—
300	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
301	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
302	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	complemented
303	C₂×C₄²	D₄×C₆₇	1	2	yes	—
304	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
305	C₂×C₄²	D₄×C₆₇	1	2	yes	—
306	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
307	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
308	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
309	#104	C₂×C₄×C₆₇	1	32	yes	direct product factor
310	#97	—	2	4	no	—
311	C₄×C₈	—	2	4	no	—
312	C₂⁴	—	2	1	no	—
313	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
314	#41	—	2	4	no	—
315	#41	—	2	8	no	—
316	#41	—	2	8	no	—
317	#41	—	2	4	no	—
318	#41	C₄²×C₆₇	1	8	yes	complemented
319	#41	C₂²×C₄×C₆₇	1	8	yes	—
320	C₂×Q₈	—	2	1	no	—
321	C₂×Q₈	—	2	2	no	—
322	C₂×Q₈	C₄²×C₆₇	1	4	yes	complemented
323	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	—	2	8	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	1	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	C₄²×C₆₇	1	4	yes	complemented
330	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
331	C₂²×C₄	—	2	16	no	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	8	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	8	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	8	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	8	no	—
341	C₂²×C₄	—	4	8	no	—
342	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
343	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
344	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
345	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
346	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
347	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
348	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
349	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
352	C₂²×C₄	#41×C₆₇	1	4	yes	—
353	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
354	M₁₆	—	2	8	no	—
355	M₁₆	—	2	4	no	—
356	M₁₆	C₂²×C₄×C₆₇	1	8	yes	complemented
357	C₂×C₈	—	2	8	no	—
358	C₂×C₈	—	2	4	no	—
359	C₄⋊C₄	—	2	8	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
362	C₂²⋊C₄	—	2	2	no	—
363	C₂²⋊C₄	—	2	2	no	—
364	C₂²⋊C₄	—	2	8	no	—
365	C₄²	—	2	32	no	—
366	C₄²	—	2	2	no	—
367	C₄²	—	2	8	no	—
368	C₄²	—	2	8	no	—
369	C₄²	—	2	2	no	—
370	C₄²	—	2	4	no	—
371	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
372	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
373	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
374	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
375	C₄²	C₂²×C₄×C₆₇	1	8	yes	complemented
376	C₄²	C₂×D₄×C₆₇	1	2	yes	—
377	C₄²	C₂×D₄×C₆₇	1	2	yes	—
378	C₂³	—	2	4	no	—
379	C₂³	—	2	2	no	—
380	C₂³	—	2	4	no	—
381	C₂³	—	4	4	no	—
382	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
383	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
384	C₂³	C₄×D₄×C₆₇	1	4	yes	—
385	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
386	Q₈	—	2	2	no	—
387	Q₈	—	2	4	no	—
388	Q₈	C₂×C₄²×C₆₇	1	4	yes	complemented
389	D₄	—	2	8	no	—
390	D₄	—	2	8	no	—
391	D₄	—	2	2	no	—
392	D₄	—	2	4	no	—
393	D₄	C₂×C₄²×C₆₇	1	4	yes	complemented
394	C₂×C₄	—	2	16	no	—
395	C₂×C₄	—	2	32	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	8	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	8	no	—
401	C₂×C₄	—	2	8	no	—
402	C₂×C₄	—	2	8	no	—
403	C₂×C₄	—	2	8	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	4	4	no	—
406	C₂×C₄	—	4	16	no	—
407	C₂×C₄	#104×C₆₇	1	8	yes	direct product factor
408	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
409	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
410	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
411	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
412	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
413	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
414	C₂×C₄	#117×C₆₇	1	2	yes	—
415	C₂×C₄	#117×C₆₇	1	2	yes	—
416	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
417	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
418	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
419	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
420	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
421	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
422	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
423	C₈	—	2	8	no	—
424	C₂²	—	2	8	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	2	2	no	—
427	C₂²	—	2	4	no	—
428	C₂²	—	4	8	no	—
429	C₂²	—	4	4	no	—
430	C₂²	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
431	C₂²	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
432	C₂²	C₂×#104×C₆₇	1	4	yes	—
433	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
434	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
435	C₄	—	2	16	no	—
436	C₄	—	2	4	no	—
437	C₄	—	2	4	no	—
438	C₄	—	4	4	no	—
439	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
440	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
441	C₄	C₂×#104×C₆₇	1	8	yes	direct product factor
442	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
443	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
444	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
445	C₂	—	2	2	no	—
446	C₂	—	2	2	no	—
447	C₂	—	4	4	no	—
448	C₂	C₄×#104×C₆₇	1	4	yes	direct product factor
449	C₂	C₂×C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
450	C₂	C₂²×#104×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₆₇	D₄
commutator	C₄	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
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. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
4. C₂×C₄³×C₆₇	C₂	semidirect
5. C₄×#104×C₆₇	C₂	direct (N × H)
6. C₂×#339×C₆₇	C₂	semidirect
11. C₄×#41×C₆₇	C₄	semidirect
13. C₂×C₄×Q₈×C₆₇	C₄	semidirect
15. C₂×C₄×D₄×C₆₇	C₄	semidirect
22. C₂×#104×C₆₇	C₄	direct (N × H)
25. C₄×M₁₆×C₆₇	C₂²	semidirect
29. C₄³×C₆₇	C₂²	semidirect
30. #339×C₆₇	C₂²	semidirect
47. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
51. C₄×Q₈×C₆₇	C₂×C₄	semidirect
55. C₄×D₄×C₆₇	C₂×C₄	semidirect
76. C₂×C₄²×C₆₇	C₂×C₄	semidirect
83. #104×C₆₇	C₂×C₄	direct (N × H)
92. #41×C₆₇	C₄²	semidirect
96. C₂×Q₈×C₆₇	C₄²	semidirect
103. C₂×D₄×C₆₇	C₄²	semidirect
130. M₁₆×C₆₇	C₂²×C₄	semidirect
149. C₄²×C₆₇	C₂²×C₄	semidirect
162. Q₈×C₆₇	C₂×C₄²	semidirect
167. D₄×C₆₇	C₂×C₄²	semidirect
181. C₂×C₄×C₆₇	#104	direct (N × H)
215. C₄×C₆₇	C₂×#104	direct (N × H)
219. C₂×C₄×#104	C₆₇	direct (N × H)
223. C₂×C₆₇	C₄×#104	direct (N × H)
228. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
229. C₂×C₄³	C₂×C₆₇	semidirect
230. C₄×#104	C₂×C₆₇	direct (N × H)
231. C₂×#339	C₂×C₆₇	semidirect
232. C₆₇	C₂×C₄×#104	direct (N × H)
237. C₄×#41	C₄×C₆₇	semidirect
239. C₂×C₄×Q₈	C₄×C₆₇	semidirect
241. C₂×C₄×D₄	C₄×C₆₇	semidirect
248. C₂×#104	C₄×C₆₇	direct (N × H)
251. C₄×M₁₆	C₂²×C₆₇	semidirect
255. C₄³	C₂²×C₆₇	semidirect
256. #339	C₂²×C₆₇	semidirect
273. C₂×M₁₆	C₂×C₄×C₆₇	semidirect
277. C₄×Q₈	C₂×C₄×C₆₇	semidirect
281. C₄×D₄	C₂×C₄×C₆₇	semidirect
302. C₂×C₄²	C₂×C₄×C₆₇	semidirect
309. #104	C₂×C₄×C₆₇	direct (N × H)
318. #41	C₄²×C₆₇	semidirect
322. C₂×Q₈	C₄²×C₆₇	semidirect
329. C₂×D₄	C₄²×C₆₇	semidirect
356. M₁₆	C₂²×C₄×C₆₇	semidirect
375. C₄²	C₂²×C₄×C₆₇	semidirect
388. Q₈	C₂×C₄²×C₆₇	semidirect
393. D₄	C₂×C₄²×C₆₇	semidirect
407. C₂×C₄	#104×C₆₇	direct (N × H)
441. C₄	C₂×#104×C₆₇	direct (N × H)
448. C₂	C₄×#104×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
C₂²×#104×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₂×C₄³×C₆₇	2
C₄×#104×C₆₇	2
C₂×#339×C₆₇	2
C₂×C₄×#104	67

Subgroups — C2×C4×#104×C67 #30699653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#104×C₆₇ #30699653