Subgroups — C₆₇×#21240 #30700491

Order 17152.

1466 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#421×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2879	C₂	1	2	yes	maximal, complemented
5	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1975	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1974	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₆₇	C₂²	1	2	yes	—
9	C₂×#135×C₆₇	—	2	4	no	—
10	C₂×#135×C₆₇	—	2	4	no	—
11	C₂×#135×C₆₇	C₄	1	2	yes	complemented
12	C₂×#135×C₆₇	C₄	1	2	yes	complemented
13	C₂×#135×C₆₇	C₂²	1	2	yes	—
14	C₂×#135×C₆₇	C₂²	1	2	yes	—
15	C₂²×Q₁₆×C₆₇	C₄	1	1	yes	characteristic, complemented
16	C₂²×SD₁₆×C₆₇	C₄	1	2	yes	complemented
17	C₂²×D₈×C₆₇	C₄	1	1	yes	characteristic, complemented
18	C₂²×M₁₆×C₆₇	C₂²	1	2	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	C₂×#108×C₆₇	—	2	4	no	—
23	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
24	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
25	#421×C₆₇	—	2	8	no	—
26	C₂×#104×C₆₇	C₂²	1	8	yes	complemented
27	#408×C₆₇	—	2	4	no	—
28	#408×C₆₇	C₂²	1	8	yes	complemented
29	#407×C₆₇	—	2	4	no	—
30	#405×C₆₇	—	2	4	no	—
31	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
32	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
33	C₂×#41×C₆₇	—	2	2	no	—
34	C₂×#41×C₆₇	—	2	2	no	—
35	C₂×#41×C₆₇	—	2	4	no	—
36	C₂×#41×C₆₇	—	2	4	no	—
37	C₂×#41×C₆₇	—	2	4	no	—
38	C₂×#41×C₆₇	—	2	4	no	—
39	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
40	C₂×#41×C₆₇	C₂³	1	4	yes	—
41	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
42	C₂²×D₄×C₆₇	—	2	2	no	—
43	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
44	C₂³×C₄×C₆₇	—	2	2	no	—
45	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	#135×C₆₇	—	2	8	no	—
47	#135×C₆₇	—	2	8	no	—
48	#135×C₆₇	—	2	4	no	—
49	#135×C₆₇	—	2	4	no	—
50	#135×C₆₇	—	2	4	no	—
51	#135×C₆₇	—	2	4	no	—
52	C₂×Q₁₆×C₆₇	—	2	4	no	—
53	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
54	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×SD₁₆×C₆₇	—	2	8	no	—
56	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
57	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
58	C₂×D₈×C₆₇	—	2	4	no	—
59	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×M₁₆×C₆₇	—	2	2	no	—
62	C₂×M₁₆×C₆₇	—	2	2	no	—
63	C₂×M₁₆×C₆₇	—	2	2	no	—
64	C₂×M₁₆×C₆₇	—	2	4	no	—
65	C₂×M₁₆×C₆₇	—	2	4	no	—
66	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
67	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
68	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
69	C₂²×C₈×C₆₇	—	2	2	no	—
70	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₈×C₆₇	D₄	1	4	yes	—
75	C₂²×C₈×C₆₇	D₄	1	4	yes	—
76	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
77	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	#117×C₆₇	—	2	2	no	—
80	#117×C₆₇	D₄	1	4	yes	—
81	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
82	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
83	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
84	#108×C₆₇	—	2	4	no	—
85	#108×C₆₇	—	2	4	no	—
86	#104×C₆₇	—	2	16	no	—
87	C₂²⋊C₈×C₆₇	—	2	16	no	—
88	#97×C₆₇	—	2	4	no	—
89	C₄×C₈×C₆₇	—	2	4	no	—
90	C₂⁴×C₆₇	—	2	2	no	—
91	C₂⁴×C₆₇	#41	1	1	yes	characteristic
92	#41×C₆₇	—	2	8	no	—
93	#41×C₆₇	—	2	8	no	—
94	#41×C₆₇	—	2	4	no	—
95	#41×C₆₇	—	2	4	no	—
96	#41×C₆₇	—	2	4	no	—
97	#41×C₆₇	—	2	4	no	—
98	#41×C₆₇	—	4	4	no	—
99	#41×C₆₇	—	4	4	no	—
100	#41×C₆₇	—	4	4	no	—
101	#41×C₆₇	—	4	4	no	—
102	C₂×Q₈×C₆₇	—	2	2	no	—
103	C₂×Q₈×C₆₇	—	2	4	no	—
104	C₂×Q₈×C₆₇	—	2	4	no	—
105	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
106	C₂×D₄×C₆₇	—	2	4	no	—
107	C₂×D₄×C₆₇	—	2	4	no	—
108	C₂×D₄×C₆₇	—	2	4	no	—
109	C₂×D₄×C₆₇	—	2	4	no	—
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	4	4	no	—
112	C₂×D₄×C₆₇	—	4	4	no	—
113	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	4	no	—
117	C₂²×C₄×C₆₇	—	2	4	no	—
118	C₂²×C₄×C₆₇	—	4	4	no	—
119	C₂²×C₄×C₆₇	—	4	4	no	—
120	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	#41	1	2	yes	—
129	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
130	Q₁₆×C₆₇	—	2	4	no	—
131	Q₁₆×C₆₇	—	2	4	no	—
132	SD₁₆×C₆₇	—	2	8	no	—
133	SD₁₆×C₆₇	—	2	8	no	—
134	D₈×C₆₇	—	2	4	no	—
135	D₈×C₆₇	—	2	4	no	—
136	M₁₆×C₆₇	—	2	4	no	—
137	M₁₆×C₆₇	—	2	4	no	—
138	M₁₆×C₆₇	—	2	4	no	—
139	M₁₆×C₆₇	—	2	4	no	—
140	M₁₆×C₆₇	—	4	4	no	—
141	C₂×C₈×C₆₇	—	2	2	no	—
142	C₂×C₈×C₆₇	—	2	2	no	—
143	C₂×C₈×C₆₇	—	2	4	no	—
144	C₂×C₈×C₆₇	—	4	4	no	—
145	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
146	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
147	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
148	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
149	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
154	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
155	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
156	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
157	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
158	C₄⋊C₄×C₆₇	—	2	4	no	—
159	C₂²⋊C₄×C₆₇	—	2	4	no	—
160	C₄²×C₆₇	—	2	2	no	—
161	C₄²×C₆₇	C₂×D₄	1	4	yes	—
162	C₂³×C₆₇	—	2	4	no	—
163	C₂³×C₆₇	—	4	2	no	—
164	C₂³×C₆₇	—	4	4	no	—
165	C₂³×C₆₇	—	8	4	no	—
166	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
167	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
168	C₂³×C₆₇	#121	1	2	yes	—
169	C₂³×C₆₇	#124	1	2	yes	—
170	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
171	Q₈×C₆₇	—	2	4	no	—
172	Q₈×C₆₇	—	2	4	no	—
173	Q₈×C₆₇	—	4	4	no	—
174	D₄×C₆₇	—	2	4	no	—
175	D₄×C₆₇	—	2	4	no	—
176	D₄×C₆₇	—	4	4	no	—
177	D₄×C₆₇	—	4	4	no	—
178	D₄×C₆₇	—	4	8	no	—
179	D₄×C₆₇	—	4	4	no	—
180	C₂×C₄×C₆₇	—	2	4	no	—
181	C₂×C₄×C₆₇	—	2	4	no	—
182	C₂×C₄×C₆₇	—	4	4	no	—
183	C₂×C₄×C₆₇	—	4	4	no	—
184	C₂×C₄×C₆₇	—	4	4	no	—
185	C₂×C₄×C₆₇	—	4	4	no	—
186	C₂×C₄×C₆₇	—	4	8	no	—
187	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
188	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
189	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
190	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
191	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	#121	1	2	yes	—
194	C₂×C₄×C₆₇	#121	1	2	yes	—
195	C₂×C₄×C₆₇	#121	1	2	yes	—
196	C₂×C₄×C₆₇	#124	1	2	yes	—
197	C₂×C₄×C₆₇	#127	1	2	yes	—
198	C₂×C₄×C₆₇	#127	1	2	yes	—
199	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
205	C₈×C₆₇	—	2	4	no	—
206	C₈×C₆₇	—	4	4	no	—
207	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
208	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
209	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
210	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
211	C₂²×C₆₇	—	2	2	no	—
212	C₂²×C₆₇	—	2	2	no	—
213	C₂²×C₆₇	—	2	2	no	—
214	C₂²×C₆₇	—	4	2	no	—
215	C₂²×C₆₇	—	4	2	no	—
216	C₂²×C₆₇	—	4	8	no	—
217	C₂²×C₆₇	—	4	4	no	—
218	C₂²×C₆₇	—	8	8	no	—
219	C₂²×C₆₇	#390	1	2	yes	—
220	C₂²×C₆₇	#390	1	2	yes	—
221	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
223	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
224	C₄×C₆₇	—	4	4	no	—
225	C₄×C₆₇	—	4	4	no	—
226	C₄×C₆₇	#390	1	2	yes	—
227	C₄×C₆₇	#390	1	2	yes	—
228	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
229	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
230	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
231	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
232	#21240	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
233	C₂×C₆₇	—	2	1	no	—
234	C₂×C₆₇	—	2	2	no	—
235	C₂×C₆₇	—	2	1	no	—
236	C₂×C₆₇	—	2	2	no	—
237	C₂×C₆₇	—	8	4	no	—
238	C₂×C₆₇	C₂×#390	1	1	yes	characteristic
239	C₂×C₆₇	#2957	1	2	yes	—
240	C₂²×#135	C₂×C₆₇	1	1	yes	characteristic
241	C₂²×#108	C₂×C₆₇	1	1	yes	characteristic, complemented
242	C₂×#421	C₂×C₆₇	1	2	yes	complemented
243	#2879	C₂×C₆₇	1	2	yes	complemented
244	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
245	#1975	C₂×C₆₇	1	4	yes	complemented
246	#1974	C₂×C₆₇	1	4	yes	complemented
247	C₆₇	#21240	1	1	yes	characteristic, direct product factor, Sylow
248	C₂²×#41	C₂²×C₆₇	1	2	yes	—
249	C₂×#135	—	2	4	no	—
250	C₂×#135	—	2	4	no	—
251	C₂×#135	C₄×C₆₇	1	2	yes	complemented
252	C₂×#135	C₄×C₆₇	1	2	yes	complemented
253	C₂×#135	C₂²×C₆₇	1	2	yes	—
254	C₂×#135	C₂²×C₆₇	1	2	yes	—
255	C₂²×Q₁₆	C₄×C₆₇	1	1	yes	characteristic, complemented
256	C₂²×SD₁₆	C₄×C₆₇	1	2	yes	complemented
257	C₂²×D₈	C₄×C₆₇	1	1	yes	characteristic, complemented
258	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
259	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
260	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
261	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
262	C₂×#108	—	2	4	no	—
263	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
264	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
265	#421	—	2	8	no	—
266	C₂×#104	C₂²×C₆₇	1	8	yes	complemented
267	#408	—	2	4	no	—
268	#408	C₂²×C₆₇	1	8	yes	complemented
269	#407	—	2	4	no	—
270	#405	—	2	4	no	—
271	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
272	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
273	C₂×#41	—	2	2	no	—
274	C₂×#41	—	2	2	no	—
275	C₂×#41	—	2	4	no	—
276	C₂×#41	—	2	4	no	—
277	C₂×#41	—	2	4	no	—
278	C₂×#41	—	2	4	no	—
279	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
280	C₂×#41	C₂³×C₆₇	1	4	yes	—
281	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
282	C₂²×D₄	—	2	2	no	—
283	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
284	C₂³×C₄	—	2	2	no	—
285	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
286	#135	—	2	8	no	—
287	#135	—	2	8	no	—
288	#135	—	2	4	no	—
289	#135	—	2	4	no	—
290	#135	—	2	4	no	—
291	#135	—	2	4	no	—
292	C₂×Q₁₆	—	2	4	no	—
293	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
294	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
295	C₂×SD₁₆	—	2	8	no	—
296	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	—
297	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	—
298	C₂×D₈	—	2	4	no	—
299	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
300	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
301	C₂×M₁₆	—	2	2	no	—
302	C₂×M₁₆	—	2	2	no	—
303	C₂×M₁₆	—	2	2	no	—
304	C₂×M₁₆	—	2	4	no	—
305	C₂×M₁₆	—	2	4	no	—
306	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
307	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
308	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
309	C₂²×C₈	—	2	2	no	—
310	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
311	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
312	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
313	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
314	C₂²×C₈	D₄×C₆₇	1	4	yes	—
315	C₂²×C₈	D₄×C₆₇	1	4	yes	—
316	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
317	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
318	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
319	#117	—	2	2	no	—
320	#117	D₄×C₆₇	1	4	yes	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₂²⋊C₄	—	2	1	no	—
323	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
324	#108	—	2	4	no	—
325	#108	—	2	4	no	—
326	#104	—	2	16	no	—
327	C₂²⋊C₈	—	2	16	no	—
328	#97	—	2	4	no	—
329	C₄×C₈	—	2	4	no	—
330	C₂⁴	—	2	2	no	—
331	C₂⁴	#41×C₆₇	1	1	yes	characteristic
332	#41	—	2	8	no	—
333	#41	—	2	8	no	—
334	#41	—	2	4	no	—
335	#41	—	2	4	no	—
336	#41	—	2	4	no	—
337	#41	—	2	4	no	—
338	#41	—	4	4	no	—
339	#41	—	4	4	no	—
340	#41	—	4	4	no	—
341	#41	—	4	4	no	—
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	2	4	no	—
344	C₂×Q₈	—	2	4	no	—
345	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
346	C₂×D₄	—	2	4	no	—
347	C₂×D₄	—	2	4	no	—
348	C₂×D₄	—	2	4	no	—
349	C₂×D₄	—	2	4	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	4	4	no	—
352	C₂×D₄	—	4	4	no	—
353	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	4	4	no	—
359	C₂²×C₄	—	4	4	no	—
360	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
361	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
363	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
364	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
365	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
366	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
367	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
368	C₂²×C₄	#41×C₆₇	1	2	yes	—
369	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
370	Q₁₆	—	2	4	no	—
371	Q₁₆	—	2	4	no	—
372	SD₁₆	—	2	8	no	—
373	SD₁₆	—	2	8	no	—
374	D₈	—	2	4	no	—
375	D₈	—	2	4	no	—
376	M₁₆	—	2	4	no	—
377	M₁₆	—	2	4	no	—
378	M₁₆	—	2	4	no	—
379	M₁₆	—	2	4	no	—
380	M₁₆	—	4	4	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	4	no	—
384	C₂×C₈	—	4	4	no	—
385	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
386	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
387	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
388	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
389	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
390	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
391	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
392	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
393	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
394	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
395	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
396	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
397	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
398	C₄⋊C₄	—	2	4	no	—
399	C₂²⋊C₄	—	2	4	no	—
400	C₄²	—	2	2	no	—
401	C₄²	C₂×D₄×C₆₇	1	4	yes	—
402	C₂³	—	2	4	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	4	4	no	—
405	C₂³	—	8	4	no	—
406	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
407	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
408	C₂³	#121×C₆₇	1	2	yes	—
409	C₂³	#124×C₆₇	1	2	yes	—
410	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
411	Q₈	—	2	4	no	—
412	Q₈	—	2	4	no	—
413	Q₈	—	4	4	no	—
414	D₄	—	2	4	no	—
415	D₄	—	2	4	no	—
416	D₄	—	4	4	no	—
417	D₄	—	4	4	no	—
418	D₄	—	4	8	no	—
419	D₄	—	4	4	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	4	no	—
422	C₂×C₄	—	4	4	no	—
423	C₂×C₄	—	4	4	no	—
424	C₂×C₄	—	4	4	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	8	no	—
427	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
428	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
429	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
430	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
431	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
432	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
433	C₂×C₄	#121×C₆₇	1	2	yes	—
434	C₂×C₄	#121×C₆₇	1	2	yes	—
435	C₂×C₄	#121×C₆₇	1	2	yes	—
436	C₂×C₄	#124×C₆₇	1	2	yes	—
437	C₂×C₄	#127×C₆₇	1	2	yes	—
438	C₂×C₄	#127×C₆₇	1	2	yes	—
439	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
440	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
441	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
442	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
443	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
444	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
445	C₈	—	2	4	no	—
446	C₈	—	4	4	no	—
447	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
448	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
449	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
450	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
451	C₂²	—	2	2	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	2	no	—
454	C₂²	—	4	2	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	4	4	no	—
457	C₂²	—	4	8	no	—
458	C₂²	—	8	8	no	—
459	C₂²	#390×C₆₇	1	2	yes	—
460	C₂²	#390×C₆₇	1	2	yes	—
461	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
462	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
463	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
464	C₄	—	4	4	no	—
465	C₄	—	4	4	no	—
466	C₄	#390×C₆₇	1	2	yes	—
467	C₄	#390×C₆₇	1	2	yes	—
468	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
469	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
470	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
471	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
472	C₂	—	2	1	no	—
473	C₂	—	2	2	no	—
474	C₂	—	2	1	no	—
475	C₂	—	2	2	no	—
476	C₂	—	8	4	no	—
477	C₂	C₂×#390×C₆₇	1	1	yes	characteristic
478	C₂	C₆₇×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×#135×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#108×C₆₇	C₂	semidirect
3. C₂×#421×C₆₇	C₂	semidirect
4. C₆₇×#2879	C₂	semidirect
5. C₂×#405×C₆₇	C₂	semidirect
6. C₆₇×#1975	C₂	semidirect
7. C₆₇×#1974	C₂	semidirect
11. C₂×#135×C₆₇	C₄	semidirect
12. C₂×#135×C₆₇	C₄	semidirect
15. C₂²×Q₁₆×C₆₇	C₄	semidirect
16. C₂²×SD₁₆×C₆₇	C₄	semidirect
17. C₂²×D₈×C₆₇	C₄	semidirect
23. C₂×#108×C₆₇	C₂²	semidirect
24. C₂×#108×C₆₇	C₂²	semidirect
26. C₂×#104×C₆₇	C₂²	semidirect
28. #408×C₆₇	C₂²	semidirect
31. C₂×#97×C₆₇	C₂²	semidirect
32. C₂×C₄×C₈×C₆₇	C₂²	semidirect
232. #21240	C₆₇	direct (N × H)
241. C₂²×#108	C₂×C₆₇	semidirect
242. C₂×#421	C₂×C₆₇	semidirect
243. #2879	C₂×C₆₇	semidirect
244. C₂×#405	C₂×C₆₇	semidirect
245. #1975	C₂×C₆₇	semidirect
246. #1974	C₂×C₆₇	semidirect
247. C₆₇	#21240	direct (N × H)
251. C₂×#135	C₄×C₆₇	semidirect
252. C₂×#135	C₄×C₆₇	semidirect
255. C₂²×Q₁₆	C₄×C₆₇	semidirect
256. C₂²×SD₁₆	C₄×C₆₇	semidirect
257. C₂²×D₈	C₄×C₆₇	semidirect
263. C₂×#108	C₂²×C₆₇	semidirect
264. C₂×#108	C₂²×C₆₇	semidirect
266. C₂×#104	C₂²×C₆₇	semidirect
268. #408	C₂²×C₆₇	semidirect
271. C₂×#97	C₂²×C₆₇	semidirect
272. C₂×C₄×C₈	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#135×C₆₇	2
C₂²×#108×C₆₇	2
C₂×#421×C₆₇	2
C₆₇×#2879	2
C₂×#405×C₆₇	2
C₆₇×#1975	2
C₆₇×#1974	2
#21240	67

Subgroups — C67×#21240 #30700491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21240 #30700491