Subgroups — C₂×#22560333 #30856004

Order 17152.

786 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523979	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
3	#22560333	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863167	C₂	1	2	yes	maximal
5	C₂×#16863152	C₂	1	2	yes	maximal
6	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	2	yes	—
8	C₂×#14524002	C₂²	1	2	yes	—
9	C₂×#14523979	—	2	1	no	—
10	C₂×#14523979	—	2	1	no	—
11	C₂×#14523979	—	2	2	no	—
12	C₂×#14523979	C₂²	1	4	yes	—
13	#16863434	—	2	2	no	—
14	#16863434	C₂²	1	4	yes	—
15	C₂×#14523977	—	2	1	no	—
16	C₂×#14523975	—	2	1	no	—
17	#16863167	C₂²	1	8	yes	—
18	#16863152	C₂²	1	8	yes	—
19	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
22	#405×C₆₇	C₄	1	4	yes	complemented
23	#405×C₆₇	C₂²	1	4	yes	—
24	C₂×#97×C₆₇	C₄	1	2	yes	complemented
25	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	complemented
26	C₂×#11761612	—	2	2	no	—
27	C₂×#11761612	—	2	2	no	—
28	C₂×#11761612	—	2	2	no	—
29	C₂×#11761612	—	2	2	no	—
30	C₂×#11761612	—	2	2	no	—
31	C₂×#11761612	C₂³	1	4	yes	—
32	C₂²×C₆₇⋊C₈	—	2	2	no	—
33	#14524002	—	2	2	no	—
34	#14524002	—	2	2	no	—
35	#14524002	—	2	2	no	—
36	#14524002	C₂³	1	4	yes	—
37	C₂×C₄⋊Dic₆₇	—	2	2	no	—
38	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
39	C₂×C₄×Dic₆₇	—	2	2	no	—
40	#14523979	—	2	4	no	—
41	#14523979	—	2	4	no	—
42	#14523977	—	2	4	no	—
43	#14523975	—	2	4	no	—
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₆₇	—	2	1	no	—
46	C₂×M₁₆×C₆₇	—	2	1	no	—
47	C₂×M₁₆×C₆₇	—	2	1	no	—
48	C₂×M₁₆×C₆₇	—	2	1	no	—
49	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	D₄	1	2	yes	—
58	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
62	#117×C₆₇	—	2	1	no	—
63	#117×C₆₇	—	2	1	no	—
64	#117×C₆₇	C₂×C₄	1	2	yes	—
65	#117×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
69	#97×C₆₇	C₂×C₄	1	8	yes	complemented
70	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
71	C₂²×Dic₆₇	—	4	2	no	—
72	#11761612	—	2	4	no	—
73	#11761612	—	2	4	no	—
74	#11761612	—	4	2	no	—
75	#11761612	—	4	2	no	—
76	C₂×C₆₇⋊C₈	—	2	4	no	—
77	C₂×C₆₇⋊C₈	—	4	2	no	—
78	C₂×C₆₇⋊C₈	—	4	2	no	—
79	C₄⋊Dic₆₇	—	2	4	no	—
80	C₄⋊Dic₆₇	—	4	2	no	—
81	C₂²⋊Dic₆₇	—	2	4	no	—
82	C₂²⋊Dic₆₇	—	4	2	no	—
83	C₄×Dic₆₇	—	2	4	no	—
84	C₄×Dic₆₇	—	4	2	no	—
85	C₂⁴×C₆₇	#41	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
101	M₁₆×C₆₇	—	2	2	no	—
102	M₁₆×C₆₇	—	2	2	no	—
103	M₁₆×C₆₇	—	2	2	no	—
104	M₁₆×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
110	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
113	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
114	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
115	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
116	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
117	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
118	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
120	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
123	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₂²⋊C₄×C₆₇	—	2	2	no	—
127	C₂²⋊C₄×C₆₇	—	2	2	no	—
128	C₄²×C₆₇	—	2	2	no	—
129	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
130	C₂³×C₆₇	—	4	2	no	—
131	C₂³×C₆₇	#117	1	1	yes	characteristic
132	C₂³×C₆₇	#117	1	1	yes	characteristic
133	C₂³×C₆₇	#124	1	1	yes	characteristic
134	C₂³×C₆₇	#124	1	1	yes	characteristic
135	C₂³×C₆₇	#125	1	1	yes	characteristic
136	C₂³×C₆₇	#125	1	1	yes	characteristic
137	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₆₇	—	2	4	no	—
139	C₂×C₄×C₆₇	—	4	4	no	—
140	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
145	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
146	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
147	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
148	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
149	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
150	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
151	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
152	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
153	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
154	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
155	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
156	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
157	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
158	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
159	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
160	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
161	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
168	C₂×Dic₆₇	—	4	4	no	—
169	C₂×Dic₆₇	—	8	4	no	—
170	C₈×C₆₇	—	2	4	no	—
171	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
172	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
173	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
174	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
175	C₆₇⋊C₈	—	4	4	no	—
176	C₂²×C₆₇	—	2	2	no	—
177	C₂²×C₆₇	—	2	2	no	—
178	C₂²×C₆₇	—	2	2	no	—
179	C₂²×C₆₇	—	4	1	no	—
180	C₂²×C₆₇	—	4	1	no	—
181	C₂²×C₆₇	—	4	1	no	—
182	C₂²×C₆₇	—	4	1	no	—
183	C₂²×C₆₇	#382	1	1	yes	characteristic
184	C₂²×C₆₇	#382	1	1	yes	characteristic
185	C₂²×C₆₇	#382	1	1	yes	characteristic
186	C₂²×C₆₇	#382	1	1	yes	characteristic
187	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
188	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
189	C₂²×C₆₇	C₂×#125	1	1	yes	characteristic
190	C₄×C₆₇	—	4	4	no	—
191	C₄×C₆₇	#382	1	1	yes	characteristic
192	C₄×C₆₇	#382	1	1	yes	characteristic
193	C₄×C₆₇	#382	1	1	yes	characteristic
194	C₄×C₆₇	#382	1	1	yes	characteristic
195	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
196	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
197	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
198	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
199	Dic₆₇	—	8	4	no	—
200	C₂×#1833	—	67	1	no	maximal, Sylow
201	C₂×C₆₇	—	2	1	no	—
202	C₂×C₆₇	—	2	1	no	—
203	C₂×C₆₇	—	2	1	no	—
204	C₂×C₆₇	—	2	1	no	—
205	C₂×C₆₇	—	2	1	no	—
206	C₂×C₆₇	—	2	1	no	—
207	C₂×C₆₇	#1833	1	2	yes	complemented
208	C₂×C₆₇	C₂×#382	1	1	yes	characteristic
209	C₂²×#108	—	67	1	no	—
210	C₂×#427	—	67	1	no	—
211	C₂×#405	D₆₇	1	1	yes	characteristic
212	#1833	—	67	8	no	—
213	C₂×#344	—	67	2	no	—
214	C₂×H(3,ℤ/4)	—	67	2	no	—
215	C₆₇	C₂×#1833	1	1	yes	characteristic, complemented, Sylow
216	C₂²×M₁₆	—	67	2	no	—
217	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
218	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
219	C₂×#117	—	67	2	no	—
220	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
221	C₂×#108	—	67	4	no	—
222	C₂×#108	—	134	1	no	—
223	C₂×#108	—	134	1	no	—
224	C₂×#108	—	134	2	no	—
225	#427	—	67	4	no	—
226	#427	—	134	2	no	—
227	C₂×#107	—	134	1	no	—
228	C₂×#106	—	134	1	no	—
229	#405	Dic₆₇	1	4	yes	complemented
230	#405	C₂×D₆₇	1	4	yes	—
231	C₂×#97	Dic₆₇	1	2	yes	complemented
232	C₂×C₄×C₈	Dic₆₇	1	2	yes	complemented
233	#344	—	67	8	no	—
234	H(3,ℤ/4)	—	67	8	no	—
235	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
236	C₂×M₁₆	—	2	1	no	—
237	C₂×M₁₆	—	2	1	no	—
238	C₂×M₁₆	—	2	1	no	—
239	C₂×M₁₆	—	2	1	no	—
240	C₂×M₁₆	—	67	4	no	—
241	C₂×M₁₆	—	134	2	no	—
242	C₂×M₁₆	—	134	2	no	—
243	C₂×M₁₆	—	134	2	no	—
244	C₂×M₁₆	—	134	2	no	—
245	C₂×M₁₆	—	134	2	no	—
246	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
247	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
248	C₂²×C₈	—	134	2	no	—
249	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
250	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
251	C₂²×C₈	Dic₁₃₄	1	2	yes	—
252	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
253	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
254	C₂²×C₈	D₂₆₈	1	2	yes	—
255	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
256	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
257	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
258	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
259	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
260	#117	—	2	1	no	—
261	#117	—	2	1	no	—
262	#117	—	67	4	no	—
263	#117	—	134	2	no	—
264	#117	—	134	2	no	—
265	#117	—	134	2	no	—
266	#117	C₂×Dic₆₇	1	2	yes	—
267	#117	C₂²×D₆₇	1	2	yes	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	134	2	no	—
270	C₂×C₂²⋊C₄	—	2	1	no	—
271	C₂×C₂²⋊C₄	—	134	2	no	—
272	C₂×C₄²	—	134	2	no	—
273	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
274	#108	—	134	4	no	—
275	#108	—	134	4	no	—
276	#107	—	134	4	no	—
277	#106	—	134	4	no	—
278	#97	C₂×Dic₆₇	1	8	yes	complemented
279	C₄×C₈	C₂×Dic₆₇	1	8	yes	complemented
280	C₂⁴	#11761628	1	1	yes	characteristic
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	268	2	no	—
283	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
284	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
287	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
288	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
289	C₂²×C₄	#11761628	1	1	yes	characteristic
290	C₂²×C₄	#11761628	1	1	yes	characteristic
291	C₂²×C₄	#11761628	1	1	yes	characteristic
292	C₂²×C₄	#11761628	1	1	yes	characteristic
293	C₂²×C₄	#11761628	1	1	yes	characteristic
294	C₂²×C₄	#11761628	1	1	yes	characteristic
295	C₂²×C₄	#11761628	1	1	yes	characteristic
296	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
297	M₁₆	—	2	2	no	—
298	M₁₆	—	2	2	no	—
299	M₁₆	—	2	2	no	—
300	M₁₆	—	2	2	no	—
301	M₁₆	—	268	2	no	—
302	M₁₆	—	268	2	no	—
303	M₁₆	—	134	4	no	—
304	M₁₆	—	134	4	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	268	2	no	—
308	C₂×C₈	—	268	2	no	—
309	C₂×C₈	—	134	4	no	—
310	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
311	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
312	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
313	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
314	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
315	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
316	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
317	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
318	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
319	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
320	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
321	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
322	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
323	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
324	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
325	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
326	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	268	2	no	—
330	C₄⋊C₄	—	134	4	no	—
331	C₂²⋊C₄	—	2	2	no	—
332	C₂²⋊C₄	—	2	2	no	—
333	C₂²⋊C₄	—	268	2	no	—
334	C₂²⋊C₄	—	134	4	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	268	2	no	—
337	C₄²	—	134	4	no	—
338	C₄²	C₂²×Dic₆₇	1	4	yes	—
339	C₂³	—	4	2	no	—
340	C₂³	#14524002	1	1	yes	characteristic
341	C₂³	#14524002	1	1	yes	characteristic
342	C₂³	#14524031	1	1	yes	characteristic
343	C₂³	#14524031	1	1	yes	characteristic
344	C₂³	#14524035	1	1	yes	characteristic
345	C₂³	#14524035	1	1	yes	characteristic
346	C₂³	C₂×#11761628	1	1	yes	characteristic
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	4	4	no	—
349	C₂×C₄	—	268	4	no	—
350	C₂×C₄	—	536	4	no	—
351	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
352	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
353	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
354	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
355	C₂×C₄	#14524002	1	1	yes	characteristic
356	C₂×C₄	#14524002	1	1	yes	characteristic
357	C₂×C₄	#14524002	1	1	yes	characteristic
358	C₂×C₄	#14524002	1	1	yes	characteristic
359	C₂×C₄	#14524002	1	1	yes	characteristic
360	C₂×C₄	#14524002	1	1	yes	characteristic
361	C₂×C₄	#14524031	1	1	yes	characteristic
362	C₂×C₄	#14524031	1	1	yes	characteristic
363	C₂×C₄	#14524031	1	1	yes	characteristic
364	C₂×C₄	#14524031	1	1	yes	characteristic
365	C₂×C₄	#14524031	1	1	yes	characteristic
366	C₂×C₄	#14524031	1	1	yes	characteristic
367	C₂×C₄	#14524035	1	1	yes	characteristic
368	C₂×C₄	#14524035	1	1	yes	characteristic
369	C₂×C₄	#14524035	1	1	yes	characteristic
370	C₂×C₄	#14524035	1	1	yes	characteristic
371	C₂×C₄	#14524035	1	1	yes	characteristic
372	C₂×C₄	#14524035	1	1	yes	characteristic
373	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
375	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
376	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
377	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
378	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
379	C₈	—	2	4	no	—
380	C₈	—	268	4	no	—
381	C₈	C₂×C₄⋊Dic₆₇	1	2	yes	—
382	C₈	C₂×C₄⋊Dic₆₇	1	2	yes	—
383	C₈	C₂×C₄⋊Dic₆₇	1	2	yes	—
384	C₈	C₂×C₄⋊Dic₆₇	1	2	yes	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	—	2	2	no	—
388	C₂²	—	4	1	no	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	4	1	no	—
392	C₂²	#16863262	1	1	yes	characteristic
393	C₂²	#16863262	1	1	yes	characteristic
394	C₂²	#16863262	1	1	yes	characteristic
395	C₂²	#16863262	1	1	yes	characteristic
396	C₂²	C₂×#14524002	1	1	yes	characteristic
397	C₂²	C₂×#14524031	1	1	yes	characteristic
398	C₂²	C₂×#14524035	1	1	yes	characteristic
399	C₄	—	4	4	no	—
400	C₄	—	536	4	no	—
401	C₄	#16863262	1	1	yes	characteristic
402	C₄	#16863262	1	1	yes	characteristic
403	C₄	#16863262	1	1	yes	characteristic
404	C₄	#16863262	1	1	yes	characteristic
405	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
406	C₄	C₂×#14524002	1	1	yes	characteristic
407	C₄	C₂×#14524031	1	1	yes	characteristic
408	C₄	C₂×#14524035	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	2	1	no	—
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	2	1	no	—
415	C₂	#22560333	1	2	yes	direct product factor
416	C₂	C₂×#16863262	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524002
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#405×C₆₇	C₂
socle	C₂²×C₆₇	#382
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#405	D₆₇
hypocenter	C₆₇	C₂×#1833
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #22560333	C₂	direct (N × H)
22. #405×C₆₇	C₄	semidirect
24. C₂×#97×C₆₇	C₄	semidirect
25. C₂×C₄×C₈×C₆₇	C₄	semidirect
69. #97×C₆₇	C₂×C₄	semidirect
70. C₄×C₈×C₆₇	C₂×C₄	semidirect
207. C₂×C₆₇	#1833	semidirect
215. C₆₇	C₂×#1833	semidirect
229. #405	Dic₆₇	semidirect
231. C₂×#97	Dic₆₇	semidirect
232. C₂×C₄×C₈	Dic₆₇	semidirect
278. #97	C₂×Dic₆₇	semidirect
279. C₄×C₈	C₂×Dic₆₇	semidirect
415. C₂	#22560333	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523979	2
C₂×#16863434	2
#22560333	2
C₂×#16863167	2
C₂×#16863152	2
C₂×#405×C₆₇	2
C₂×#1833	67

Subgroups — C2×#22560333 #30856004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560333 #30856004